Roots of quadratic equations

Size: px
Start display at page:

Download "Roots of quadratic equations"

Transcription

1 CHAPTER Roots of quadratic equations Learning objectives After studying this chapter, you should: know the relationships between the sum and product of the roots of a quadratic equation and the coefficients of the equation be able to manipulate expressions involving and be able to form equations with roots related to a given quadratic equation. In this chapter you will be looking at quadratic equations with particular emphasis on the properties of their solutions or roots.. The relationships between the roots and coefficients of a quadratic equation As you have already seen in the C module, any quadratic equation will have two roots (even though one may be a repeated root or the roots may not even be real). In this section you will be considering some further properties of these two roots. Suppose you know that the two solutions of a quadratic equation are x and x 5 and you want to find a quadratic equation having and 5 as its roots. The method consists of working backwards, i.e. following the steps for solving a quadratic equation but in reverse order. Now if x and x 5 are the solutions then the equation could have been factorised as (x )(x 5) 0. Expanding the brackets gives x 3x 0 0. This is a quadratic equation with roots and 5. Actually, any multiple of this equation will also have the same roots, e.g. x 3 6x 0 0 3x 9x 30 0 x 3 x 5 0

2 Roots of quadratic equations The general case Consider the most general quadratic equation ax bx c 0 and suppose that the two solutions are x and x. Now if and are the roots of the equation then you can work backwards to generate the original equation. A quadratic with the two solutions x and x is (x )(x ) 0. Expanding the brackets gives x x x 0 x ( )x 0 [] The most general quadratic equation is ax bx c 0 and this can easily be written in the same form as equation []. You can divide ax bx c 0 throughout by a giving Any multiple of this equation such as kx k( )x k 0 will also have roots and. x b a x c 0 [] a The coefficients of x in [] and in [] are now both equal to. Since [] and [] have the same roots, and, and are in the same form, you can write x ( )x x b a x c a Equating coefficients of x gives ( ) b a b a Equating constant terms gives Recall that the coefficient of x is the number in front of x. The sign means identically equal to. The coefficients of x and the constant terms must be equal. a c When the quadratic equation ax bx c 0 has roots and : The sum of the roots, b a ; and the product of roots, a c. Notice it is fairly easy to express x b a x c 0 as a x (sum of roots)x (product of roots) 0

3 Roots of quadratic equations 3 Worked example. Write down the sum and the product of the roots for each of the following equations: (a) x x 3 0, (b) x 8x 5 0. (a) x x 3 0 Here a, b and c 3. The sum of roots, b a 6. c 3 The product of roots, a. (b) x 8x 5 0 Here a, b 8 and c 5. The sum of roots, b a 8 8. c 5 The product of roots, 5. a Worked example. Find the sum and the product of the roots of each of the following quadratic equations: (a) 4x 8x 5, (b) x(x 4) 6 x. Take careful note of the signs. Notice the double negative. Neither equation is in the form ax bx c 0 and so the first thing to do is to get them into this standard form. (a) 4x 8x 5 4x 8x 5 0 In this case a 4, b 8 and c 5. The sum of roots, b a 8. 4 c 5 The product of roots, 5 a 4 4. (b) x(x 4) 6 x Expand the brackets and take everything onto the LHS. x 4x x 6 0 x x 6 0 Here a, b and c 6. The sum of roots, b a. c 6 The product of roots, 6. a Now in the form ax bx c 0. Now in the standard form.

4 4 Roots of quadratic equations Worked example.3 Write down equations with integer coefficients for which: (a) sum of roots 4, product of roots 7, (b) sum of roots 4, product of roots 5, (c) sum of roots 3 5, product of roots. (a) A quadratic equation can be written as x (sum of roots)x (product of roots) 0 x (4)x (7) 0 x 4x 7 0 (b) Using x (sum of roots)x (product of roots) 0 gives x (4)x (5) 0 x 4x 5 0 (c) Using x (sum of roots)x (product of roots) 0 gives x ( 3 5 )x ( ) 0 x 3 5 x 0 0x 0( 3 5 )x 0( ) 0 0x 6x 5 0 Again great care must be taken with the signs. Some of the coefficients are fractions not integers. You can eliminate the fractions by multiplying throughout by 0 or ( 5). You now have integer coefficients. EXERCISE A Find the sum and product of the roots for each of the following quadratic equations: (a) x 4x 9 0 (b) x 3x 5 0 (c) x 0x 3 0 (d) x 3x 0 (e) 7x x 6 (f) x(x ) x 6 (g) x(3 x) 5x (h) ax a x a (i) ax 8a ( a)x (j) x 3 x 5 Write down a quadratic equation with: (a) sum of roots 5, product of roots 8, (b) sum of roots 3, product of roots 5, (c) sum of roots 4, product of roots 7, (d) sum of roots 9, product of roots 4, (e) sum of roots 4, product of roots 5, (f) sum of roots 3, product of roots 4, (g) sum of roots 3 5, product of roots 0, (h) sum of roots k, product of roots 3k, (i) sum of roots k, product of roots 6 k, (j) sum of roots ( a ), product of roots (a 7).

5 Roots of quadratic equations 5. Manipulating expressions involving and As you have seen in the last section, given a quadratic equation with roots and you can find the values of and without solving the equation. As you will see in this section, it is useful to be able to write other expressions involving and in terms of and. Worked example.4 Given that 4 and 7, find the values of: (a), (b). (a) You need to write this expression in terms of and in order to use the values given in the question. Now 4 7 (b) () 7 49 Two relations which will prove very useful are ( ) 3 3 ( ) 3 3( ) Notice how the expressions on the RHS contain combinations of just and. These two results can be proved fairly easily: ( + ) = ( + )( + ) ( + ) = = ( + ) as required and ( ) 3 ( )( )( ) ( ) 3 ( )( ) ( ) ( ) ( ) ( ) 3 3 ( ) 3 3( ) as required Take 3 as a factor.

6 6 Roots of quadratic equations Worked example.5 Given that 5 and, find the values of: (a), (b) 3 3, (c). (a) + = + = ( + ) Substitute the known values of and 5 () Now it is in terms of and. (b) 3 3 ( ) 3 3( ) ()(5) (c) ( ) ( ) 5 ( ( ) ) Worked example.6 Given that 5 and 3, find the value of ( ). ( ) ( ) ( ) 4 ( ) 5 4( 3 ) Worked example.7 Write the expression in terms of and. 3 3 ( )3 3( ) ()

7 Roots of quadratic equations 7 EXERCISE B Write each of the following expressions in terms of and : (a) (b) (c) 3 3 (d) (e) ( )( ) (f) 5 5 Given that 3 and 9, find the values of: (a) 3 3, (b). 3 Given that 4 and 0, find the values of: (a), (b) Given that 7 and, find the values of: (a), (b). 5 The roots of the quadratic equation x 5x 3 0 are and. (a) Write down the values of and. (b) Hence find the values of: (i) ( 3)( 3), (ii). 6 The roots of the equation x 4x 3 0 are and. Without solving the equation, find the value of: (a) (b) (c) (d) (e) ( ) (f) ( )( ) 7 The roots of the quadratic equation x 4x 0 are and. (a) Find the values of: (i), (ii). (b) Hence find the value of: (i) ( )( ), (ii)..3 Forming new equations with related roots It is often possible to find a quadratic equation whose roots are related in some way to the roots of another given quadratic equation.

8 8 Roots of quadratic equations Worked example.8 The roots of the equation x 5x 6 0 are and. Find a quadratic equation whose roots are and. x 5x 6 0 The sum of roots, b a 5 5. c 6 The product of roots, 3. a You would be able to write down the new equation with roots and if you could find the sum and the product of the new roots. The sum of new roots, 5 6. The product of new roots, 3 3. Using x (sum of roots)x (product of roots) 0 the equation with roots and is x 5 6 x x 5 6 x 0 3 6x 5x 0. The last example illustrates the basic method for forming new equations with roots that are related to the roots of a given equation: Multiply through by 6 in order to obtain integer coefficients (because it is often required in an examination question). Write down the sum of the roots,, and the product of the roots,, of the given equation. Find the sum and product of the new roots in terms of and. 3 Write down the new equation using x (sum of new roots)x (product of new roots) 0.

9 Roots of quadratic equations 9 Worked example.9 The roots of the equation x 7x 0 are and. Find the values of and. Hence, find a quadratic equation whose roots are and. From the equation x 7x 0 you have The sum of roots, b a 7 7. c The product of roots,. a Now, ( ) (7) () and () () 4. Now since the roots of the new equation are and, and are the sum and product of the new roots. The required equation is x (sum of new roots)x (product of new roots) 0 x 53x 4 0. You could use any variable you choose. For instance, you could write y 53y 4 0. Worked example.0 The roots of the quadratic equation x 5x 3 0 are and. Find a quadratic equation whose roots are 3 and 3. From the equation x 5x 3 0 you have The sum of roots, b a 5 5. c 3 The product of roots, 3. a The new equation has roots 3 and 3. The sum of new roots, 3 3 ( ) 3 3( ) (5) 3 3(3)(5) The product of new roots, 3 3 () 3 (3) 3 7. Using x (sum of new roots)x (product of new roots) 0 the required equation is x 70x 7 0.

10 0 Roots of quadratic equations EXERCISE C The roots of the equation x 6x 4 0 are and. Find a quadratic equation whose roots are and. The roots of the equation x 3x 5 0 are and. Find a quadratic equation whose roots are and. 3 The roots of the equation x 9x 5 0 are and. Find a quadratic equation whose roots are and. 4 Given that the roots of the equation x 5x 3 0 are and, find an equation with integer coefficients whose roots are and. 5 Given that the roots of the equation 3x 6x 0 are and, find a quadratic equation with integer coefficients whose roots are 3 and 3. 6 The roots of the equation x 4x 0 are and. Find a quadratic equation with integer coefficients whose roots are and. 7 The roots of the equation 3x 6x 0 are and. (a) Find the value of. (b) Find a quadratic equation whose roots are and. 8 The roots of the equation x 7x 3 0 are and. Without solving this equation, (a) find the value of 3 3. (b) Hence, find a quadratic equation with integer coefficients which has roots and. 9 Given that and are the roots of the equation 5x x 4 0, find a quadratic equation with integer coefficients which has roots and. 0 The roots of the equation x 4x 6 0 are and. Find an equation whose roots are and..4 Further examples Sometimes the questions may require you to apply similar techniques to the previous section but without directing you to find the sum and product of the roots.

11 Roots of quadratic equations Worked example. The roots of the equation x kx 8 0 are and 3. Find the two possible values of k. From x kx 8 0 you have The sum of roots, ( 3) k 3 k [] The product of roots, ( + 3) = = 8 [] Equations [] and [] form a pair of simultaneous equations. From [] you have k 3 and substituting this into [] gives k 3 3 k 3 8 k 6k 9 3k k 6k 9 6k 8 k the two possible values for k are and..5 New equations by means of a substitution Worked example.8 is reproduced below. The roots of the equation x 5x 6 0 are and. Find a quadratic equation whose roots are and. Since the new equation has roots that are the reciprocals of the original equation, the new equation can be found very quickly by making the substitution y x x, which transforms y x 5x 6 0 into y y Multiplying throughout by y gives 5y 6y 0 or 6y 5y 0. Check the working in Worked example.8 to see which is quicker.

12 Roots of quadratic equations Worked example. The roots of the equation x 3x 5 0 are and. Find quadratic equations with roots (a) 3 and 3, (b) and. (a) Use the substitution y x 3 in x 3x 5 0, since the roots in the new equation are 3 less than those in the original equation. y x 3 x y 3, so new equation is (y 3) 3(y 3) 5 0 or y 6y 9 3y The new equation is y 3y 5 0. (b) This time you need to use the substitution y x in x 3x 5 0 to eliminate x. Hence, y 3x 5 0 or y 5 3x. Squaring both sides gives (y 5) 9x 9y. Hence, y 0y 5 9y or y 9y 5 0. You could solve this question using the sum and product of roots technique of the previous section if you prefer. If you prefer, you can write the new equation as x 3x 5 0 or in terms of any other variable. Alternative solution Using 3 and 5 (a) Sum of new roots is ( 3) ( 3) 6 3 Product of new roots is ( 3)( 3) ( ) 9 5 Hence new equation is y 3y 5 0. (b) Sum of new roots is ( ) Product of new roots is () 5 Hence new equation is y 9y 5 0. You should see that the answers are the same whichever method you use. If the question directs you to use a particular substitution, e.g. Use the substitution y x to find a new equation whose roots are and, you must use the first method. However, if the question is of the form Use the substitution y x, or otherwise, to find a new equation whose roots are and, or simply Find an equation whose roots are and you may use either technique.

13 Roots of quadratic equations 3 Worked examination question.3 The roots of the quadratic equation x 3x 7 0 are and. (a) Write down the values of (i), (ii). (b) Find a quadratic equation with integer coefficients whose roots are and. (a) From the equation x 3x 7 0 you have The sum of roots, b a 3 3. c 7 The product of roots, 7. a (b) The new equation has roots and. The sum of new roots, ( ) 3 (7) The product of new roots,. Using x (sum of new roots)x (product of new roots) 0 the required equation is x 3 x 0 7 x 3 x 0 7 7x 3x 7 0 A common mistake is to forget to multiply throughout by 7 to obtain integer coefficients. EXERCISE D The roots of the equation x 9x k 0 are and. Find the value of k. The roots of the quadratic equation 4x kx 5 0 are and 3. Find the two possible values of the constant k. 3 The roots of the quadratic equation x 3x 7 0 are and. Find an equation with integer coefficients which has roots 3 3 and. 4 The roots of the quadratic equation x 5x 0 are and. Find an equation with integer coefficients which has roots and.

14 4 Roots of quadratic equations 5 The roots of the quadratic equation x x 5 0 are and. Find a quadratic equation which has roots and. 6 The roots of the quadratic equation x 5x 7 0 are and. (a) Without solving the equation, find the values of (i), (ii) 3 3. (b) Determine an equation with integer coefficients which has roots and. [A] 7 The roots of the quadratic equation 3x 4x 0 are and. (a) Without solving the equation, find the values of (i), (ii) 3 3. (b) Determine a quadratic equation with integer coefficients which has roots 3 and 3. [A] 8 The roots of the quadratic equation x x 3 0 are and. (a) Without solving the equation: (i) write down the value of and the value of, (ii) show that 3 3 0, (iii)find the value of 3 3. (b) Determine a quadratic equation with integer coefficients which has roots 3 and. [A] 3 9 The roots of the quadratic equation x 3x 0 are and. (a) Without solving the equation: (i) show that 7, (ii) find the value of 3 3. (b) (i) Show that 4 4 ( ) (). (ii) Hence, find the value of 4 4. (c) Determine a quadratic equation with integer coefficients which has roots ( 3 ) and ( 3 ). [A] 0 The roots of the quadratic equation x 3x 0 are and. (a) Write down the values of and. (b) Without solving the equation, find the value of: (i), (ii) 3 3. (c) Determine a quadratic equation with integer coefficients 3 which has roots and. [A] 3

15 Roots of quadratic equations 5 Key point summary When the quadratic equation ax bx c 0 has roots and : p The sum of the roots, b a ; and the product of roots, c. a A quadratic equation can be expressed as x (sum of roots)x (product of roots) 0. p 3 Two useful results are: p5 ( ) 3 3 ( ) 3 3( ). 4 The basic method for forming new equations with roots that are related to the roots of p8 a given equation is: Write down the sum of the roots,, and the product of the roots,, of the given equation. Find the sum and product of the new roots in terms of and. 3 Write down the new equation using x (sum of new roots)x (product of new roots) 0 Test yourself Find the sums and products of the roots of the following Section. equations: (a) x 6x 4 0, (b) x x 5 0. Write down the equations, with integer coefficients, where: Section.3 (a) sum of roots 4, product of roots 7, (b) sum of roots 5 4, product of roots. What to review 3 The roots of the equation x 5x 6 0 are and. Section. Find the values of: (a) + (b). 4 Given that the roots of 3x 9x 0 are and, find Section.3 a quadratic equation whose roots are and. Test yourself ANSWERS (a) sum = 6, product = 4; (b) sum, product = 5. (a) x 4x 7 0; (b) 4x 5x 0. 3 (a) (b) 4 x 7x ; 3. 6

Factoring Quadratic Expressions

Factoring Quadratic Expressions Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the

More information

Factoring Trinomials of the Form x 2 bx c

Factoring Trinomials of the Form x 2 bx c 4.2 Factoring Trinomials of the Form x 2 bx c 4.2 OBJECTIVES 1. Factor a trinomial of the form x 2 bx c 2. Factor a trinomial containing a common factor NOTE The process used to factor here is frequently

More information

Mathematics Higher Tier, Algebraic Fractions

Mathematics Higher Tier, Algebraic Fractions These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are a school or an organisation and would like to purchase these solutions please contact Chatterton

More information

Factorising quadratics

Factorising quadratics Factorising quadratics An essential skill in many applications is the ability to factorise quadratic expressions. In this unit you will see that this can be thought of as reversing the process used to

More information

Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if

Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if 1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c

More information

Algebra Revision Sheet Questions 2 and 3 of Paper 1

Algebra Revision Sheet Questions 2 and 3 of Paper 1 Algebra Revision Sheet Questions and of Paper Simple Equations Step Get rid of brackets or fractions Step Take the x s to one side of the equals sign and the numbers to the other (remember to change the

More information

Key. Introduction. What is a Quadratic Equation? Better Math Numeracy Basics Algebra - Rearranging and Solving Quadratic Equations.

Key. Introduction. What is a Quadratic Equation? Better Math Numeracy Basics Algebra - Rearranging and Solving Quadratic Equations. Key On screen content Narration voice-over Activity Under the Activities heading of the online program Introduction This topic will cover: the definition of a quadratic equation; how to solve a quadratic

More information

Solving Quadratic Equations by Factoring

Solving Quadratic Equations by Factoring 4.7 Solving Quadratic Equations by Factoring 4.7 OBJECTIVE 1. Solve quadratic equations by factoring The factoring techniques you have learned provide us with tools for solving equations that can be written

More information

Polynomials and Vieta s Formulas

Polynomials and Vieta s Formulas Polynomials and Vieta s Formulas Misha Lavrov ARML Practice 2/9/2014 Review problems 1 If a 0 = 0 and a n = 3a n 1 + 2, find a 100. 2 If b 0 = 0 and b n = n 2 b n 1, find b 100. Review problems 1 If a

More information

I N D U BRIDGING THE GAP TO A/S LEVEL MATHS C T I O N

I N D U BRIDGING THE GAP TO A/S LEVEL MATHS C T I O N I N D U BRIDGING THE GAP TO A/S LEVEL MATHS C T I O N INTRODUCTION TO A LEVEL MATHS The Mathematics Department is committed to ensuring that you make good progress throughout your A level or AS course.

More information

FACTORING QUADRATIC EQUATIONS

FACTORING QUADRATIC EQUATIONS FACTORING QUADRATIC EQUATIONS Summary 1. Difference of squares... 1 2. Mise en évidence simple... 2 3. compounded factorization... 3 4. Exercises... 7 The goal of this section is to summarize the methods

More information

SOLVING EQUATIONS. Firstly, we map what has been done to the variable (in this case, xx):

SOLVING EQUATIONS. Firstly, we map what has been done to the variable (in this case, xx): CONNECT: Algebra SOLVING EQUATIONS An Algebraic equation is where two algebraic expressions are equal to each other. Finding the solution of the equation means finding the value(s) of the variable which

More information

Expanding brackets and factorising

Expanding brackets and factorising Chapter 7 Expanding brackets and factorising This chapter will show you how to expand and simplify expressions with brackets solve equations and inequalities involving brackets factorise by removing a

More information

3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes

3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same

More information

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

More information

STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS

STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an

More information

Step 1: Set the equation equal to zero if the function lacks. Step 2: Subtract the constant term from both sides:

Step 1: Set the equation equal to zero if the function lacks. Step 2: Subtract the constant term from both sides: In most situations the quadratic equations such as: x 2 + 8x + 5, can be solved (factored) through the quadratic formula if factoring it out seems too hard. However, some of these problems may be solved

More information

is identically equal to x 2 +3x +2

is identically equal to x 2 +3x +2 Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3

More information

Level 2 Certificate Further MATHEMATICS

Level 2 Certificate Further MATHEMATICS Level 2 Certificate Further MATHEMATICS 83601 Paper 1 non-calculator Report on the Examination Specification 8360 June 2013 Version: 1.0 Further copies of this Report are available from aqa.org.uk Copyright

More information

Mathematics Higher Tier, Simultaneous Equations

Mathematics Higher Tier, Simultaneous Equations These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are a school or an organisation and would like to purchase these solutions please contact Chatterton

More information

Maths Module 6 Algebra Solving Equations This module covers concepts such as:

Maths Module 6 Algebra Solving Equations This module covers concepts such as: Maths Module 6 Algebra Solving Equations This module covers concepts such as: solving equations with variables on both sides multiples and factors factorising and expanding function notation www.jcu.edu.au/students/learning-centre

More information

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with

More information

Year 9 set 1 Mathematics notes, to accompany the 9H book.

Year 9 set 1 Mathematics notes, to accompany the 9H book. Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H

More information

QUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE

QUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE MODULE - 1 Quadratic Equations 6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write

More information

The Method of Partial Fractions Math 121 Calculus II Spring 2015

The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

More information

You should aim to complete the booklet and mymaths task for enrolment. You can bring this with you to enrolment and also to your first lesson.

You should aim to complete the booklet and mymaths task for enrolment. You can bring this with you to enrolment and also to your first lesson. Pre-enrolment task for 2014 entry Mathematics Why do I need to complete a pre-enrolment task? To be successful on AS Maths it is vital that you are able to use higher level GCSE topics as soon as you start

More information

Generating Functions

Generating Functions Generating Functions If you take f(x =/( x x 2 and expand it as a power series by long division, say, then you get f(x =/( x x 2 =+x+2x 2 +x +5x 4 +8x 5 +x 6 +. It certainly seems as though the coefficient

More information

5.4 The Quadratic Formula

5.4 The Quadratic Formula Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

More information

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

More information

LESSON 1 PRIME NUMBERS AND FACTORISATION

LESSON 1 PRIME NUMBERS AND FACTORISATION LESSON 1 PRIME NUMBERS AND FACTORISATION 1.1 FACTORS: The natural numbers are the numbers 1,, 3, 4,. The integers are the naturals numbers together with 0 and the negative integers. That is the integers

More information

Introduction to polynomials

Introduction to polynomials Worksheet 4.5 Polynomials Section 1 Introduction to polynomials A polynomial is an expression of the form p(x) = p 0 + p 1 x + p 2 x 2 + + p n x n, (n N) where p 0, p 1,..., p n are constants and x os

More information

Lesson 3.2 Exercises, pages

Lesson 3.2 Exercises, pages Lesson 3.2 Exercises, pages 190 195 Students should verify all the solutions. A 4. Which equations are quadratic equations? Explain how you know. a) 3x 2 = 30 b) x 2-9x + 8 = 0 This equation is quadratic

More information

Equations, Inequalities & Partial Fractions

Equations, Inequalities & Partial Fractions Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities

More information

The Not-Formula Book for C1

The Not-Formula Book for C1 Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes

More information

LINEAR EQUATIONS 7YEARS. A guide for teachers - Years 7 8 June The Improving Mathematics Education in Schools (TIMES) Project

LINEAR EQUATIONS 7YEARS. A guide for teachers - Years 7 8 June The Improving Mathematics Education in Schools (TIMES) Project LINEAR EQUATIONS NUMBER AND ALGEBRA Module 26 A guide for teachers - Years 7 8 June 2011 7YEARS 8 Linear Equations (Number and Algebra : Module 26) For teachers of Primary and Secondary Mathematics 510

More information

1.1 Solving a Linear Equation ax + b = 0

1.1 Solving a Linear Equation ax + b = 0 1.1 Solving a Linear Equation ax + b = 0 To solve an equation ax + b = 0 : (i) move b to the other side (subtract b from both sides) (ii) divide both sides by a Example: Solve x = 0 (i) x- = 0 x = (ii)

More information

is identically equal to x 2 +3x +2

is identically equal to x 2 +3x +2 Partial fractions.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as + for any

More information

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P. MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

More information

First Degree Equations First degree equations contain variable terms to the first power and constants.

First Degree Equations First degree equations contain variable terms to the first power and constants. Section 4 7: Solving 2nd Degree Equations First Degree Equations First degree equations contain variable terms to the first power and constants. 2x 6 = 14 2x + 3 = 4x 15 First Degree Equations are solved

More information

FACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1

FACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1 5.7 Factoring ax 2 bx c (5-49) 305 5.7 FACTORING ax 2 bx c In this section In Section 5.5 you learned to factor certain special polynomials. In this section you will learn to factor general quadratic polynomials.

More information

Solving Rational Equations

Solving Rational Equations Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

More information

Mth 95 Module 2 Spring 2014

Mth 95 Module 2 Spring 2014 Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression

More information

MATHEMATICS FOR ENGINEERING BASIC ALGEBRA

MATHEMATICS FOR ENGINEERING BASIC ALGEBRA MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL 3 EQUATIONS This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals.

More information

4. Changing the Subject of a Formula

4. Changing the Subject of a Formula 4. Changing the Subject of a Formula This booklet belongs to aquinas college maths dept. 2010 1 Aims and Objectives Changing the subject of a formula is the same process as solving equations, in that we

More information

1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes

1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.

More information

Teaching Quadratic Functions

Teaching Quadratic Functions Teaching Quadratic Functions Contents Overview Lesson Plans Worksheets A1 to A12: Classroom activities and exercises S1 to S11: Homework worksheets Teaching Quadratic Functions: National Curriculum Content

More information

FACTORING QUADRATICS 8.1.1 and 8.1.2

FACTORING QUADRATICS 8.1.1 and 8.1.2 FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.

More information

To add fractions we rewrite the fractions with a common denominator then add the numerators. = +

To add fractions we rewrite the fractions with a common denominator then add the numerators. = + Partial Fractions Adding fractions To add fractions we rewrite the fractions with a common denominator then add the numerators. Example Find the sum of 3 x 5 The common denominator of 3 and x 5 is 3 x

More information

Name Intro to Algebra 2. Unit 1: Polynomials and Factoring

Name Intro to Algebra 2. Unit 1: Polynomials and Factoring Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332

More information

Equations and Inequalities

Equations and Inequalities Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

More information

Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:

Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra: Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than

More information

not to be republished NCERT QUADRATIC EQUATIONS CHAPTER 4 (A) Main Concepts and Results

not to be republished NCERT QUADRATIC EQUATIONS CHAPTER 4 (A) Main Concepts and Results QUADRATIC EQUATIONS (A) Main Concepts and Results Quadratic equation : A quadratic equation in the variable x is of the form ax + bx + c = 0, where a, b, c are real numbers and a 0. Roots of a quadratic

More information

Integrating algebraic fractions

Integrating algebraic fractions Integrating algebraic fractions Sometimes the integral of an algebraic fraction can be found by first epressing the algebraic fraction as the sum of its partial fractions. In this unit we will illustrate

More information

Summer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2

Summer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2 Summer Math Packet Student Name: Say Hello to Algebra 2 For Students Entering Algebra 2 This summer math booklet was developed to provide students in middle school an opportunity to review grade level

More information

Worksheet 4.7. Polynomials. Section 1. Introduction to Polynomials. A polynomial is an expression of the form

Worksheet 4.7. Polynomials. Section 1. Introduction to Polynomials. A polynomial is an expression of the form Worksheet 4.7 Polynomials Section 1 Introduction to Polynomials A polynomial is an expression of the form p(x) = p 0 + p 1 x + p 2 x 2 + + p n x n (n N) where p 0, p 1,..., p n are constants and x is a

More information

Class XI Chapter 5 Complex Numbers and Quadratic Equations Maths. Exercise 5.1. Page 1 of 34

Class XI Chapter 5 Complex Numbers and Quadratic Equations Maths. Exercise 5.1. Page 1 of 34 Question 1: Exercise 5.1 Express the given complex number in the form a + ib: Question 2: Express the given complex number in the form a + ib: i 9 + i 19 Question 3: Express the given complex number in

More information

( ) FACTORING. x In this polynomial the only variable in common to all is x.

( ) FACTORING. x In this polynomial the only variable in common to all is x. FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated

More information

Solving Systems by Elimination 35

Solving Systems by Elimination 35 10/21/13 Solving Systems by Elimination 35 EXAMPLE: 5x + 2y = 1 x 3y = 7 1.Multiply the Top equation by the coefficient of the x on the bottom equation and write that equation next to the first equation

More information

Lecture 7 : Inequalities 2.5

Lecture 7 : Inequalities 2.5 3 Lecture 7 : Inequalities.5 Sometimes a problem may require us to find all numbers which satisfy an inequality. An inequality is written like an equation, except the equals sign is replaced by one of

More information

3. Polynomials. 3.1 Parabola primer

3. Polynomials. 3.1 Parabola primer 3. Polynomials 3.1 Parabola primer A parabola is the graph of a quadratic (degree 2) polynomial, that is, a parabola is polynomial of the form: standard form: y = ax 2 + bx + c. When we are given a polynomial

More information

1 Lecture: Integration of rational functions by decomposition

1 Lecture: Integration of rational functions by decomposition Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

More information

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives 6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

More information

Factoring Polynomials

Factoring Polynomials Factoring Polynomials 4-1-2014 The opposite of multiplying polynomials is factoring. Why would you want to factor a polynomial? Let p(x) be a polynomial. p(c) = 0 is equivalent to x c dividing p(x). Recall

More information

Partial Fractions Examples

Partial Fractions Examples Partial Fractions Examples Partial fractions is the name given to a technique of integration that may be used to integrate any ratio of polynomials. A ratio of polynomials is called a rational function.

More information

Mathematics. (www.tiwariacademy.com : Focus on free Education) (Chapter 5) (Complex Numbers and Quadratic Equations) (Class XI)

Mathematics. (www.tiwariacademy.com : Focus on free Education) (Chapter 5) (Complex Numbers and Quadratic Equations) (Class XI) ( : Focus on free Education) Miscellaneous Exercise on chapter 5 Question 1: Evaluate: Answer 1: 1 ( : Focus on free Education) Question 2: For any two complex numbers z1 and z2, prove that Re (z1z2) =

More information

MATH REVIEW KIT. Reproduced with permission of the Certified General Accountant Association of Canada.

MATH REVIEW KIT. Reproduced with permission of the Certified General Accountant Association of Canada. MATH REVIEW KIT Reproduced with permission of the Certified General Accountant Association of Canada. Copyright 00 by the Certified General Accountant Association of Canada and the UBC Real Estate Division.

More information

Tool 1. Greatest Common Factor (GCF)

Tool 1. Greatest Common Factor (GCF) Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When

More information

GCSE MATHEMATICS. 43602H Unit 2: Number and Algebra (Higher) Report on the Examination. Specification 4360 November 2014. Version: 1.

GCSE MATHEMATICS. 43602H Unit 2: Number and Algebra (Higher) Report on the Examination. Specification 4360 November 2014. Version: 1. GCSE MATHEMATICS 43602H Unit 2: Number and Algebra (Higher) Report on the Examination Specification 4360 November 2014 Version: 1.0 Further copies of this Report are available from aqa.org.uk Copyright

More information

NSM100 Introduction to Algebra Chapter 5 Notes Factoring

NSM100 Introduction to Algebra Chapter 5 Notes Factoring Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

More information

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4) ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

More information

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

More information

Teaching & Learning Plans. Quadratic Equations. Junior Certificate Syllabus

Teaching & Learning Plans. Quadratic Equations. Junior Certificate Syllabus Teaching & Learning Plans Quadratic Equations Junior Certificate Syllabus The Teaching & Learning Plans are structured as follows: Aims outline what the lesson, or series of lessons, hopes to achieve.

More information

SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen

SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen A. GENERALITIES. When a given quadratic equation can be factored, there are 2 best methods

More information

Factoring and Applications

Factoring and Applications Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the

More information

Chapter R.4 Factoring Polynomials

Chapter R.4 Factoring Polynomials Chapter R.4 Factoring Polynomials Introduction to Factoring To factor an expression means to write the expression as a product of two or more factors. Sample Problem: Factor each expression. a. 15 b. x

More information

Algebra Tiles Activity 1: Adding Integers

Algebra Tiles Activity 1: Adding Integers Algebra Tiles Activity 1: Adding Integers NY Standards: 7/8.PS.6,7; 7/8.CN.1; 7/8.R.1; 7.N.13 We are going to use positive (yellow) and negative (red) tiles to discover the rules for adding and subtracting

More information

Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00

Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 18.781 Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list

More information

1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x).

1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x). .7. PRTIL FRCTIONS 3.7. Partial Fractions.7.. Rational Functions and Partial Fractions. rational function is a quotient of two polynomials: R(x) = P (x) Q(x). Here we discuss how to integrate rational

More information

A fairly quick tempo of solutions discussions can be kept during the arithmetic problems.

A fairly quick tempo of solutions discussions can be kept during the arithmetic problems. Distributivity and related number tricks Notes: No calculators are to be used Each group of exercises is preceded by a short discussion of the concepts involved and one or two examples to be worked out

More information

1.3 Algebraic Expressions

1.3 Algebraic Expressions 1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

More information

3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS 3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

More information

What's Going On? Quiz Tomorrow. What's My Vertex? Find my Vertex! What's my y Intercept?

What's Going On? Quiz Tomorrow. What's My Vertex? Find my Vertex! What's my y Intercept? What's Going On? Quiz Tomorrow What's My Vertex? Find my Vertex! What's my y Intercept? Learning Goal I will understand how to find the zeros or x intercepts of a quadratic equation. 1 What's my Vertex?

More information

Partial Fractions. (x 1)(x 2 + 1)

Partial Fractions. (x 1)(x 2 + 1) Partial Fractions Adding rational functions involves finding a common denominator, rewriting each fraction so that it has that denominator, then adding. For example, 3x x 1 3x(x 1) (x + 1)(x 1) + 1(x +

More information

Date: Section P.2: Exponents and Radicals. Properties of Exponents: Example #1: Simplify. a.) 3 4. b.) 2. c.) 3 4. d.) Example #2: Simplify. b.) a.

Date: Section P.2: Exponents and Radicals. Properties of Exponents: Example #1: Simplify. a.) 3 4. b.) 2. c.) 3 4. d.) Example #2: Simplify. b.) a. Properties of Exponents: Section P.2: Exponents and Radicals Date: Example #1: Simplify. a.) 3 4 b.) 2 c.) 34 d.) Example #2: Simplify. a.) b.) c.) d.) 1 Square Root: Principal n th Root: Example #3: Simplify.

More information

Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

More information

QUADRATIC EQUATIONS. 4.1 Introduction

QUADRATIC EQUATIONS. 4.1 Introduction 70 MATHEMATICS QUADRATIC EQUATIONS 4 4. Introduction In Chapter, you have studied different types of polynomials. One type was the quadratic polynomial of the form ax + bx + c, a 0. When we equate this

More information

Factoring Trinomials: The ac Method

Factoring Trinomials: The ac Method 6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For

More information

Sect 6.7 - Solving Equations Using the Zero Product Rule

Sect 6.7 - Solving Equations Using the Zero Product Rule Sect 6.7 - Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred

More information

SOLVING QUADRATIC EQUATIONS BY THE NEW TRANSFORMING METHOD (By Nghi H Nguyen Updated Oct 28, 2014))

SOLVING QUADRATIC EQUATIONS BY THE NEW TRANSFORMING METHOD (By Nghi H Nguyen Updated Oct 28, 2014)) SOLVING QUADRATIC EQUATIONS BY THE NEW TRANSFORMING METHOD (By Nghi H Nguyen Updated Oct 28, 2014)) There are so far 8 most common methods to solve quadratic equations in standard form ax² + bx + c = 0.

More information

Algebra C/D Borderline Revision Document

Algebra C/D Borderline Revision Document Algebra C/D Borderline Revision Document Algebra content that could be on Exam Paper 1 or Exam Paper 2 but: If you need to use a calculator for *** shown (Q8 and Q9) then do so. Questions are numbered

More information

Functions and Equations

Functions and Equations Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

More information

SOLVING POLYNOMIAL EQUATIONS

SOLVING POLYNOMIAL EQUATIONS C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

More information

Using the ac Method to Factor

Using the ac Method to Factor 4.6 Using the ac Method to Factor 4.6 OBJECTIVES 1. Use the ac test to determine factorability 2. Use the results of the ac test 3. Completely factor a trinomial In Sections 4.2 and 4.3 we used the trial-and-error

More information

3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes

3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

More information

The x-intercepts of the graph are the x-values for the points where the graph intersects the x-axis. A parabola may have one, two, or no x-intercepts.

The x-intercepts of the graph are the x-values for the points where the graph intersects the x-axis. A parabola may have one, two, or no x-intercepts. Chapter 10-1 Identify Quadratics and their graphs A parabola is the graph of a quadratic function. A quadratic function is a function that can be written in the form, f(x) = ax 2 + bx + c, a 0 or y = ax

More information

Session 3 Solving Linear and Quadratic Equations and Absolute Value Equations

Session 3 Solving Linear and Quadratic Equations and Absolute Value Equations Session 3 Solving Linear and Quadratic Equations and Absolute Value Equations 1 Solving Equations An equation is a statement expressing the equality of two mathematical expressions. It may have numeric

More information

Solving Quadratic Equations by Completing the Square

Solving Quadratic Equations by Completing the Square 9. Solving Quadratic Equations by Completing the Square 9. OBJECTIVES 1. Solve a quadratic equation by the square root method. Solve a quadratic equation by completing the square. Solve a geometric application

More information

REVISED GCSE Scheme of Work Mathematics Higher Unit T3. For First Teaching September 2010 For First Examination Summer 2011

REVISED GCSE Scheme of Work Mathematics Higher Unit T3. For First Teaching September 2010 For First Examination Summer 2011 REVISED GCSE Scheme of Work Mathematics Higher Unit T3 For First Teaching September 2010 For First Examination Summer 2011 Version 1: 28 April 10 Version 1: 28 April 10 Unit T3 Unit T3 This is a working

More information

Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is

Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.

More information

1 Algebra - Partial Fractions

1 Algebra - Partial Fractions 1 Algebra - Partial Fractions Definition 1.1 A polynomial P (x) in x is a function that may be written in the form P (x) a n x n + a n 1 x n 1 +... + a 1 x + a 0. a n 0 and n is the degree of the polynomial,

More information