Sample Problems. Lecture Notes Equations with Parameters page 1

Size: px
Start display at page:

Download "Sample Problems. Lecture Notes Equations with Parameters page 1"

Transcription

1 Lecture Notes Equations with Parameters page Sample Problems. In each of the parametric equations given, nd the value of the parameter m so that the equation has exactly one real solution. a) x + mx m 0 b) mx + mx + m x + c) x + m x. Consider the parametric equation 5m 3x 5mx + 6m + x 0. a) Find all values of m for which x 7 is a solution of the equation. b) Find all values of m for which there are two di erent real solutions of this equation. c) Find all values of the parameter m for which the two solutions of the equation add up to 3. d) Find all values of the parameter m for which the product of the two solutions of the equation is Consider the equation x + mx + 3mx + 5x. a) Find all values of m for which the equation has exactly one real solution. b) Find all values of m for which the equation has no real solution. c) Find all values of m for which the two real solutions x and x are such that x + x :. Consider the parametric equation m + mx + mx x a) Find all values of m for which x is a solution of the equation. b) Find all values of m for which there are two real solutions of the equation. c) Find all values of m for which there are two real solutions, x and x of the equation such that x + x 5. c copyright Hidegkuti, Powell, 00 Last revised: February 5, 009

2 Lecture Notes Equations with Parameters page Sample Problems Answers. a) 0; 6 b) 0; c) 0;. a) ; b) m 6 5 c) d) ; a) 5 ; 3; 7 b) (3; 7) c) 3. a) 0 b) ( ; 0) [ 0; c) Sample Problems Solutions. In each of the parametric equations given, nd the value of the parameter m so that the equation has exactly one solution. a) x + mx m 0 0; 6 Solution : Completing the square. x + mx + m {z } x + m x + mx m 0 m m For exactly one solution, we need that m m 0 m 0 m x + m x + mx + m m 0. We solve this equation for m. m 0 multiply by m + 6m 0 m (m + 6) 0 m 0 m 6 Solution : The Quadratic Formula. Based on our equation, we have a, b m, and c m. For exactly one solution, we need the discriminant, b ac to be zero. We solve this equation for m. b ac 0 m () ( m) 0 m + 6m 0 m (m + 6) 0 m 0 m 6 c copyright Hidegkuti, Powell, 00 Last revised: February 5, 009

3 Lecture Notes Equations with Parameters page 3 b) mx + mx + m x + 0; Solution : Completing the square. Before we proceed as usual, we have to consider the case for which the equation is linear, namely, m 0. (Remember, we can NOT treat linear equations as a special case of a quadratic equation.) If m 0, our equation is 0 x + which indeed has exactly one solution. Now, if m 6 0; we complete the square mx + x (m ) + m 0 m x + m m x + m 0 m! m x + m (m ) (m ) x + m m m + m 0 m m x + m! (m ) m m + m 0 m To have exactly one solution, the part after the complete square must be equal to zero. (m ) m + m 0 multiply by m m (m ) + m (m ) 0 m + m + m m 0 m 0 m m Solution : The Quadratic Formula. Before we do as usual, we have to consider the case for which the equation is linear, namely, m 0. (Remember, we can NOT treat linear equations as a special case of the quadratic...) If m 0, our equation is 0 x + which indeed has exactly one solution. Now, if m 6 0; we have mx + x (m ) + m 0 and thus a m, b m, and c m. To have exactly one solution, we need the discriminant, b ac to be zero. We solve this equation for m. b ac 0 (m ) (m) (m ) 0 m m + m + m 0 m + 0 m m c copyright Hidegkuti, Powell, 00 Last revised: February 5, 009

4 Lecture Notes Equations with Parameters page c) x + m x 0; Solution : Completing the square. First we separately consider the case for which the equation is linear, namely, m 0. If m 0, our equation is x which indeed has exactly one solution. Now, if m 6 0; we multiply both sides by x and complete the square x + m x multiply by x x + m x subtract x x x + m 0 factor out x x + m 0 x x x + 6! x x + {z 6} 6 + m 0! x 6 + m 0 To have exactly one solution, the part after the complete square must be equal to zero. 6 + m 0 multiply by 6 + m 0 m m There is one more possibility we need to consider. The original equation is x+ m x and NOT x x+m 0. We might get exactly one solution as follows: the discriminant is positive, indicating two solutions, but one of the solutions is 0 which would have to be ruled out as it is not in the domain of the original equation. To check out this case, we take x x + m 0 and see what m is so that one of the solutions for x is zero. x x + m 0 (0) 0 + m 0 m 0 We have already considered this case, and so our solution, m 0, m is complete. Solution : Quadratic Formula First we separately consider the case for which the equation is linear, namely, m 0. If m 0, our equation is x which indeed has exactly one solution. Now, if m 6 0; we multiply both sides by x and use the formula: x + m x multiply by x x + m x subtract x x x + m 0 Now a, b, and c m: To have exactly one solution, the discriminant needs to be zero. We solve this equation for m. b ac 0 m ( ) () m 0 m m 0 c copyright Hidegkuti, Powell, 00 Last revised: February 5, 009

5 Lecture Notes Equations with Parameters page 5 There is one more possibility we need to consider. The original equation is x + m and NOT x x x + m 0. We might get exactly one solution as follows: the discriminant is positive, indicating two solutions, but one of the solutions is 0 which would have to be ruled out as it is not in the domain of the original equation. To check out this case, we take x x + m 0 and see what m is so that one of the solutions for x is zero. x x + m 0 (0) 0 + m 0 m 0 We have already considered this case, and so our solution, m 0, m is complete.. Consider the parametric equation 5m 3x 5mx + 6m + x 0. a) Find all values of m for which x 7 is a solution of the equation. ; Solution: Just plug in x 7 into the equation and see what that gives us for m. 5m 3x 5mx + 6m + x 0 5m 3 (7) 5m (7) + 6m + (7) 0 5m 35m + 6m m 30m + 0 factor out 6 6 m 5m (m ) (m ) 0 m m b) Find all values of m for which there are two di erent solutions of this equation. m 6 5 Solution: We rearrange the polynomial by degrees of the variable x. x + x ( 5m 3) + 6m + 5m 0. Now a, b 5m 3, and c 6m + 5m. For two solutions, we need the discriminant, b ac to be positive. b ac > 0 ( 5m 3) () 6m + 5m > 0 5m + 30m + 9 m 0m + 6 > 0 m + 0m + 5 > 0 (m + 5) > 0 m 6 5 c) Find all values of the parameter m for which the two solutions of the equation add up to 3. Solution: the sum of the two solutions is b a. b a 3 ( 5m 3) 3 5m m 0 m c copyright Hidegkuti, Powell, 00 Last revised: February 5, 009

6 Lecture Notes Equations with Parameters page 6 d) Find all values of the parameter m for which the product of the two solutions of the equation is 0. ; 7 6 Solution: the product of the two solutions is c. Thus we have a c a 0 We solve for m using the quadratic formula: m ; m b p b a ac 5 p m + 5m 0 6m + 5m 0 6m + 5m 0 5 p 5 (6) ( ) (6) 5 p p 5 (6) ( ) m Consider the equation x + mx + 3mx + 5x. 5 a) Find all values of m for which the equation has exactly one real solution. 3 ; 3; 7 Solution: We rearrange the equation and obtain (3m 5) x + (m + ) x + 0. Thus a 3m 5; b m + ; and c. Case. There might be one solution if the equation is linear, i.e. if a 0. This happens when 3m 5 0 3m 5 m 5 3 Then the equation becomes. we write m 5 3 : 3 x + 0 subtract 3 x divide by 3 x 3 Thus m 5 3 gives us one solution. Case. If a 6 0, (when m 6 5 ) then the equation is quadratic. It will have exactly one solution when the 3 discriminant, b ac is zero. This will give us an equation in m. b ac (m + ) () (3m 5) m + m + m + 0 m 0m + (m 3) (m 7) m 3 m 7 c copyright Hidegkuti, Powell, 00 Last revised: February 5, 009

7 Lecture Notes Equations with Parameters page 7 Let us check one of these values. If m 3; then the equation is (3 (3) 5) x + (3 + ) x + 0 x + x + 0 (x + ) 0 x and so we have exactly one real solution. Thus there is exactly one solution for x if m 5, 3, or 7. 3 b) Find all values of m for which the equation has no real solution. (3; 7) Solution: there is no solution for a quadratic equation when the discriminant is negative. From the previous part, we have that the discriminant, as a function of m is D (m) (m 3) (m 7) This expression is clearly a quadratic polynomial, with a positive leading coe ecient. regular parabola, which is negative between the two x intercepts. Thus Thus its graph is a (m 3) (m 7) < 0 if and only if 3 < m < 7 y x c) Find all values of m for which the two real solutions x and x are such that x + x : Solution: x + x (x + x ) x x m + 3m 5 3m 5 b c a a (m + ) (3m 5) 3m 5 Thus we need to solve (m + ) (3m 5) 3m 5 for m Recall that in this case m and so 3m (m + ) (3m 5) 3m 5 mulitply by (3m 5) (3m 5) (m + ) (3m 5) subtract 9m 30m + 5 m + m + m + 0 9m 30m + 5 m + m + m + 0 9m 30m + 5 m 6m + 5m m 9 0 (5m 9) (m + ) 0 c copyright Hidegkuti, Powell, 00 Last revised: February 5, 009

8 Lecture Notes Equations with Parameters page m 9 m 5 Although we obtained two values for m; they will not both work. Since 3 < 9 5 < 7; the value m 9 5 falls into the interval where there are no real solutions. The only answer is m :. Consider the parametric equation m + mx + mx x a) Find all values of m for which x is a solution of the equation. 0 Solution: We simply substitute x into the equation and solve for m. m + m ( ) + m ( ) ( ) m m + m m 0 b) Find all values of m for which there are two real solutions of the equation. ( ; 0) [ 0; Solution: We have to be careful with parametric equations if the leading term contains a parameter. We always need to treat the linear case separately. Is there a value of m for which the equation is not quadratic? Clearly, if m 0. We check out this case separately. If m 0; then m + mx + mx x 0 + (0) x + (0) x x 0 x x 3 x Thus there is exactly one solution for m 0. Now, if m 6 0, the equation is quadratic. Quadratic equations have two di erent real solutions if the discriminant is positive. 0 mx + x (m ) + m + D b ac (m ) m (m + ) 6 6m D > 0 6 6m > 0 6 > 6m > m Thus it appears, the solution is all values of m that are less than. However, we do need to rule out m 0 since then the equation has only one solution. Thus the answer is m <, m 6 0, or, in interval notation: ( ; 0) [ 0;. c) Find all values of m for which there are two real solutions, x and x of the equation such that x + x 5. Solution: x + x (x + x ) x x b c a a b a (m ) (m + ) (m ) (m + ) (m ) m (m + ) m m m m m m (m ) m (m + ) m m 6m + 6 m m m m 0m + 6 m c copyright Hidegkuti, Powell, 00 Last revised: February 5, 009 c a

9 Lecture Notes Equations with Parameters page 9 m 0m + 6 m 5 m 0m + 6 5m 0 56m + 0m 6 0 7m + 5m 0 (7m ) (m + ) m 7 m Since 7 >, there is no solution for x if m, and thus it is not a solution. On the other hand, falls 7 into the range where the equation has two solutions and so it is correct. We can check: if m, then our equation is Thus a, b 6, and c. Since x + x m + mx + mx x m + ( ) x + ( ) x x b a x x x x x c and a x + x (x + x ) x x 0 x + 6x b c a a 6 ( 6) ( ) For more documents like this, visit our page at and click on Lecture Notes. questions or comments to c copyright Hidegkuti, Powell, 00 Last revised: February 5, 009

Sample Problems. Practice Problems

Sample Problems. Practice Problems Lecture Notes Quadratic Word Problems page 1 Sample Problems 1. The sum of two numbers is 31, their di erence is 41. Find these numbers.. The product of two numbers is 640. Their di erence is 1. Find these

More information

Sample Problems. Practice Problems

Sample Problems. Practice Problems Lecture Notes Partial Fractions page Sample Problems Compute each of the following integrals.. x dx. x + x (x + ) (x ) (x ) dx 8. x x dx... x (x + ) (x + ) dx x + x x dx x + x x + 6x x dx + x 6. 7. x (x

More information

Sample Problems. Practice Problems

Sample Problems. Practice Problems Lecture Notes Circles - Part page Sample Problems. Find an equation for the circle centered at (; ) with radius r = units.. Graph the equation + + = ( ).. Consider the circle ( ) + ( + ) =. Find all points

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

Method To Solve Linear, Polynomial, or Absolute Value Inequalities:

Method To Solve Linear, Polynomial, or Absolute Value Inequalities: Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with

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

Quadratic Functions and Parabolas

Quadratic Functions and Parabolas MATH 11 Quadratic Functions and Parabolas A quadratic function has the form Dr. Neal, Fall 2008 f () = a 2 + b + c where a 0. The graph of the function is a parabola that opens upward if a > 0, and opens

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

Section 1.1 Linear Equations: Slope and Equations of Lines

Section 1.1 Linear Equations: Slope and Equations of Lines Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

More information

a) x 2 8x = 25 x 2 8x + 16 = (x 4) 2 = 41 x = 4 ± 41 x + 1 = ± 6 e) x 2 = 5 c) 2x 2 + 2x 7 = 0 2x 2 + 2x = 7 x 2 + x = 7 2

a) x 2 8x = 25 x 2 8x + 16 = (x 4) 2 = 41 x = 4 ± 41 x + 1 = ± 6 e) x 2 = 5 c) 2x 2 + 2x 7 = 0 2x 2 + 2x = 7 x 2 + x = 7 2 Solving Quadratic Equations By Square Root Method Solving Quadratic Equations By Completing The Square Consider the equation x = a, which we now solve: x = a x a = 0 (x a)(x + a) = 0 x a = 0 x + a = 0

More information

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

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

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

This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

This is a square root. The number under the radical is 9. (An asterisk * means multiply.) Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

More information

Partial Fractions Decomposition

Partial Fractions Decomposition Partial Fractions Decomposition Dr. Philippe B. Laval Kennesaw State University August 6, 008 Abstract This handout describes partial fractions decomposition and how it can be used when integrating rational

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

This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide.

This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. COLLEGE ALGEBRA UNIT 2 WRITING ASSIGNMENT This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. 1) What is the

More information

QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS

QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS Content 1. Parabolas... 1 1.1. Top of a parabola... 2 1.2. Orientation of a parabola... 2 1.3. Intercept of a parabola... 3 1.4. Roots (or zeros) of a parabola...

More information

Guide to SRW Section 1.7: Solving inequalities

Guide to SRW Section 1.7: Solving inequalities Guide to SRW Section 1.7: Solving inequalities When you solve the equation x 2 = 9, the answer is written as two very simple equations: x = 3 (or) x = 3 The diagram of the solution is -6-5 -4-3 -2-1 0

More information

Lecture 5 : Solving Equations, Completing the Square, Quadratic Formula

Lecture 5 : Solving Equations, Completing the Square, Quadratic Formula Lecture 5 : Solving Equations, Completing the Square, Quadratic Formula An equation is a mathematical statement that two mathematical expressions are equal For example the statement 1 + 2 = 3 is read as

More information

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality. 8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

More information

Assessment Schedule 2013

Assessment Schedule 2013 NCEA Level Mathematics (9161) 013 page 1 of 5 Assessment Schedule 013 Mathematics with Statistics: Apply algebraic methods in solving problems (9161) Evidence Statement ONE Expected Coverage Merit Excellence

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

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

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions. Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

More information

Math Common Core Sampler Test

Math Common Core Sampler Test High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests

More information

2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: 2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

More information

Quadratic Modeling Business 10 Profits

Quadratic Modeling Business 10 Profits Quadratic Modeling Business 10 Profits In this activity, we are going to look at modeling business profits. We will allow q to represent the number of items manufactured and assume that all items that

More information

POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

More information

Objectives. By the time the student is finished with this section of the workbook, he/she should be able

Objectives. By the time the student is finished with this section of the workbook, he/she should be able QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a

More information

Sample Problems. Practice Problems

Sample Problems. Practice Problems Lecture Notes Factoring by the AC-method page 1 Sample Problems 1. Completely factor each of the following. a) 4a 2 mn 15abm 2 6abmn + 10a 2 m 2 c) 162a + 162b 2ax 4 2bx 4 e) 3a 2 5a 2 b) a 2 x 3 b 2 x

More information

The Point-Slope Form

The Point-Slope Form 7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

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

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real

More information

2.6 Exponents and Order of Operations

2.6 Exponents and Order of Operations 2.6 Exponents and Order of Operations We begin this section with exponents applied to negative numbers. The idea of applying an exponent to a negative number is identical to that of a positive number (repeated

More information

Algebra 1 Course Title

Algebra 1 Course Title Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

More information

Lecture Notes Order of Operations page 1

Lecture Notes Order of Operations page 1 Lecture Notes Order of Operations page 1 The order of operations rule is an agreement among mathematicians, it simpli es notation. P stands for parentheses, E for exponents, M and D for multiplication

More information

Chapter 8. Quadratic Equations and Functions

Chapter 8. Quadratic Equations and Functions Chapter 8. Quadratic Equations and Functions 8.1. Solve Quadratic Equations KYOTE Standards: CR 0; CA 11 In this section, we discuss solving quadratic equations by factoring, by using the square root property

More information

2013 MBA Jump Start Program

2013 MBA Jump Start Program 2013 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 2 1 Equation of

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

Determinants can be used to solve a linear system of equations using Cramer s Rule.

Determinants can be used to solve a linear system of equations using Cramer s Rule. 2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution

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

(2 4 + 9)+( 7 4) + 4 + 2

(2 4 + 9)+( 7 4) + 4 + 2 5.2 Polynomial Operations At times we ll need to perform operations with polynomials. At this level we ll just be adding, subtracting, or multiplying polynomials. Dividing polynomials will happen in future

More information

Algebra Practice Problems for Precalculus and Calculus

Algebra Practice Problems for Precalculus and Calculus Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials

More information

7.7 Solving Rational Equations

7.7 Solving Rational Equations Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate

More information

Algebra Cheat Sheets

Algebra Cheat Sheets Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

More information

Section 6.1 Factoring Expressions

Section 6.1 Factoring Expressions Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what

More information

Answer Key Building Polynomial Functions

Answer Key Building Polynomial Functions Answer Key Building Polynomial Functions 1. What is the equation of the linear function shown to the right? 2. How did you find it? y = ( 2/3)x + 2 or an equivalent form. Answers will vary. For example,

More information

Examples of Functions

Examples of Functions Examples of Functions In this document is provided examples of a variety of functions. The purpose is to convince the beginning student that functions are something quite different than polynomial equations.

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

More information

5.1 Radical Notation and Rational Exponents

5.1 Radical Notation and Rational Exponents Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

More information

MATH 65 NOTEBOOK CERTIFICATIONS

MATH 65 NOTEBOOK CERTIFICATIONS MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1

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

5.7 Literal Equations

5.7 Literal Equations 5.7 Literal Equations Now that we have learned to solve a variety of different equations (linear equations in chapter 2, polynomial equations in chapter 4, and rational equations in the last section) we

More information

Week 2 Quiz: Equations and Graphs, Functions, and Systems of Equations

Week 2 Quiz: Equations and Graphs, Functions, and Systems of Equations Week Quiz: Equations and Graphs, Functions, and Systems of Equations SGPE Summer School 014 June 4, 014 Lines: Slopes and Intercepts Question 1: Find the slope, y-intercept, and x-intercept of the following

More information

In this lesson you will learn to find zeros of polynomial functions that are not factorable.

In this lesson you will learn to find zeros of polynomial functions that are not factorable. 2.6. Rational zeros of polynomial functions. In this lesson you will learn to find zeros of polynomial functions that are not factorable. REVIEW OF PREREQUISITE CONCEPTS: A polynomial of n th degree has

More information

2.1 QUADRATIC FUNCTIONS AND MODELS. Copyright Cengage Learning. All rights reserved.

2.1 QUADRATIC FUNCTIONS AND MODELS. Copyright Cengage Learning. All rights reserved. 2.1 QUADRATIC FUNCTIONS AND MODELS Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze graphs of quadratic functions. Write quadratic functions in standard form and use the results

More information

Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}

Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)} Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in

More information

Core Maths C1. Revision Notes

Core Maths C1. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

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

7. Solving Linear Inequalities and Compound Inequalities

7. Solving Linear Inequalities and Compound Inequalities 7. Solving Linear Inequalities and Compound Inequalities Steps for solving linear inequalities are very similar to the steps for solving linear equations. The big differences are multiplying and dividing

More information

Methods to Solve Quadratic Equations

Methods to Solve Quadratic Equations Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a second-degree

More information

Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

More information

Packet 1 for Unit 2 Intercept Form of a Quadratic Function. M2 Alg 2

Packet 1 for Unit 2 Intercept Form of a Quadratic Function. M2 Alg 2 Packet 1 for Unit Intercept Form of a Quadratic Function M Alg 1 Assignment A: Graphs of Quadratic Functions in Intercept Form (Section 4.) In this lesson, you will: Determine whether a function is linear

More information

Vocabulary Words and Definitions for Algebra

Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

More information

Solving Quadratic Equations

Solving Quadratic Equations 9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

More information

Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

More information

4.1. COMPLEX NUMBERS

4.1. COMPLEX NUMBERS 4.1. COMPLEX NUMBERS What You Should Learn Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers

More information

1.6 A LIBRARY OF PARENT FUNCTIONS. Copyright Cengage Learning. All rights reserved.

1.6 A LIBRARY OF PARENT FUNCTIONS. Copyright Cengage Learning. All rights reserved. 1.6 A LIBRARY OF PARENT FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal

More information

MAT 135 Midterm Review Dugopolski Sections 2.2,2.3,2.5,2.6,3.3,3.5,4.1,4.2,5.7,5.8,6.1,6.2,6.3

MAT 135 Midterm Review Dugopolski Sections 2.2,2.3,2.5,2.6,3.3,3.5,4.1,4.2,5.7,5.8,6.1,6.2,6.3 Directions: Complete each problem and select the correct answer. NOTE: Not all topics on the midterm are represented in this review. For a complete set of review problems, please do the book-based midterm

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

Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities

Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities Solving systems of equations with inequalities When solving systems of linear equations, we are looking for the ordered pair

More information

6.1 Add & Subtract Polynomial Expression & Functions

6.1 Add & Subtract Polynomial Expression & Functions 6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic

More information

0.8 Rational Expressions and Equations

0.8 Rational Expressions and Equations 96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

More information

Basic numerical skills: EQUATIONS AND HOW TO SOLVE THEM. x + 5 = 7 2 + 5-2 = 7-2 5 + (2-2) = 7-2 5 = 5. x + 5-5 = 7-5. x + 0 = 20.

Basic numerical skills: EQUATIONS AND HOW TO SOLVE THEM. x + 5 = 7 2 + 5-2 = 7-2 5 + (2-2) = 7-2 5 = 5. x + 5-5 = 7-5. x + 0 = 20. Basic numerical skills: EQUATIONS AND HOW TO SOLVE THEM 1. Introduction (really easy) An equation represents the equivalence between two quantities. The two sides of the equation are in balance, and solving

More information

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions. Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

More information

Sample Problems. 10. 1 2 cos 2 x = tan2 x 1. 11. tan 2 = csc 2 tan 2 1. 12. sec x + tan x = cos x 13. 14. sin 4 x cos 4 x = 1 2 cos 2 x

Sample Problems. 10. 1 2 cos 2 x = tan2 x 1. 11. tan 2 = csc 2 tan 2 1. 12. sec x + tan x = cos x 13. 14. sin 4 x cos 4 x = 1 2 cos 2 x Lecture Notes Trigonometric Identities page Sample Problems Prove each of the following identities.. tan x x + sec x 2. tan x + tan x x 3. x x 3 x 4. 5. + + + x 6. 2 sec + x 2 tan x csc x tan x + cot x

More information

College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1

College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

More information

Week 1: Functions and Equations

Week 1: Functions and Equations Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.1-2.2, and Chapter

More information

Solving Inequalities Examples

Solving Inequalities Examples Solving Inequalities Examples 1. Joe and Katie are dancers. Suppose you compare their weights. You can make only one of the following statements. Joe s weight is less than Kate s weight. Joe s weight is

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

Representation of functions as power series

Representation of functions as power series Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

More information

Zeros of a Polynomial Function

Zeros of a Polynomial Function Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

More information

Linearly Independent Sets and Linearly Dependent Sets

Linearly Independent Sets and Linearly Dependent Sets These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation

More information

Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)

Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary) Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify

More information

Math 1050 Khan Academy Extra Credit Algebra Assignment

Math 1050 Khan Academy Extra Credit Algebra Assignment Math 1050 Khan Academy Extra Credit Algebra Assignment KhanAcademy.org offers over 2,700 instructional videos, including hundreds of videos teaching algebra concepts, and corresponding problem sets. In

More information

Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

More information

Chapter 6 Notes. Section 6.1 Solving One-Step Linear Inequalities

Chapter 6 Notes. Section 6.1 Solving One-Step Linear Inequalities Chapter 6 Notes Name Section 6.1 Solving One-Step Linear Inequalities Graph of a linear Inequality- the set of all points on a number line that represent all solutions of the inequality > or < or circle

More information

Solutions of Linear Equations in One Variable

Solutions of Linear Equations in One Variable 2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

More information

Solving Logarithmic Equations

Solving Logarithmic Equations Solving Logarithmic Equations Deciding How to Solve Logarithmic Equation When asked to solve a logarithmic equation such as log (x + 7) = or log (7x + ) = log (x + 9), the first thing we need to decide

More information

Algebra 1-2. A. Identify and translate variables and expressions.

Algebra 1-2. A. Identify and translate variables and expressions. St. Mary's College High School Algebra 1-2 The Language of Algebra What is a variable? A. Identify and translate variables and expressions. The following apply to all the skills How is a variable used

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

BookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line

BookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina - Beaufort Lisa S. Yocco, Georgia Southern University

More information

Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year.

Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Goal The goal of the summer math program is to help students

More information

Section 2.1 Intercepts; Symmetry; Graphing Key Equations

Section 2.1 Intercepts; Symmetry; Graphing Key Equations Intercepts: An intercept is the point at which a graph crosses or touches the coordinate axes. x intercept is 1. The point where the line crosses (or intercepts) the x-axis. 2. The x-coordinate of a point

More information

Math 115 Spring 2011 Written Homework 5 Solutions

Math 115 Spring 2011 Written Homework 5 Solutions . Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence

More information

Systems of Equations Involving Circles and Lines

Systems of Equations Involving Circles and Lines Name: Systems of Equations Involving Circles and Lines Date: In this lesson, we will be solving two new types of Systems of Equations. Systems of Equations Involving a Circle and a Line Solving a system

More information

Midterm 1. Solutions

Midterm 1. Solutions Stony Brook University Introduction to Calculus Mathematics Department MAT 13, Fall 01 J. Viro October 17th, 01 Midterm 1. Solutions 1 (6pt). Under each picture state whether it is the graph of a function

More information

Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test

Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action

More information

Chapter 8 Graphs and Functions:

Chapter 8 Graphs and Functions: Chapter 8 Graphs and Functions: Cartesian axes, coordinates and points 8.1 Pictorially we plot points and graphs in a plane (flat space) using a set of Cartesian axes traditionally called the x and y axes

More information