Algebra Module A6 Quadratic Equations - Copyright This publication The Northern Alberta Institute of Technology. All Rights Reserved. LAST REVISED November, 8
Quadratic Equations - Statement of Prerequisite Skills Complete all previous TLM modules before completing this module. Required Supporting Materials Access to the World Wide Web. Internet Explorer 5.5 or greater. Macromedia Flash Player. Rationale Why is it important for you to learn this material? The quadratic equation is one of the most common equations encountered in the technologies. Its applications include projectile motion, architectural problems, electrical circuits, problems in mechanics, and design. This unit will introduce the student to many different strategies for manipulating and solving quadratic equations. Learning Outcome When you complete this module you will be able to Apply additional theory on quadratic equations. Learning Objectives 1. Determine the nature of the roots of a quadratic equation by examining the value of the discriminant.. Determine the quadratic equation given the roots of the equation. 3. Determine the quadratic equation given the sum and product of the roots. Connection Activity Recall the quadratic equation: b± b 4ac x = a We used this formula to solve for the roots of a quadratic equation in previous modules. A close examination of the discriminant ( b 4ac) will yield some useful information about the quadratic in question. This module will guide you through using parts of a quadratic such as the discriminant and the roots to provide information about quadratics. Module A6 More on Quadratics - 1
OBJECTIVE ONE When you complete this objective you will be able to Determine the nature of the roots of a quadratic equation by examining the value of the discriminant. Exploration Activity Discriminant b± b 4ac In the quadratic formula x =, the quantity b 4ac is called the a discriminant. b 4ac helps us determine the type of roots we will get when we solve a quadratic equation. The discriminant does not include the radical sign. We shall shorten the word discriminant to the word disc for convenience. There are three types of discriminant that we are going to study. TYPE I: If b 4ac>, the roots are real and unequal. Example 1 Determine the discriminant for the following quadratic equation. Then use this to solve the equation. x x 3= In this equation a = 1, b =, c = 3. Substituting into b 4ac we get: b 4ac = ( ) 4(1)( 3) = 16 Since 16 >, i.e, the discriminant >, the roots are real and unequal. Knowing this, solve the quadratic: x x 3= (x 3)(x + 1) = x = 3, and x = 1 Module A6 More on Quadratics -
Check in x x 3= by substituting. For x = 3: 3 (3) 3 = = so 3 works. For x = 1: ( 1) ( 1) 3 = = so 1 works. TYPE II If b 4ac=, the roots are real and equal. Example Determine the discriminant for the following quadratic equation. Then use this to solve the equation. 4x + 8x + 49 = Here a = 4, b = 8, c = 49 b 4ac= 8 4(4)(49) = Since the discriminant =, the roots are real and equal. Knowing that the roots are real and equal makes the equation easier to factor: 4x + 8x + 49 = (x + 7)(x + 7) = 7 7 x =, You should check these answers by substitution. TYPE III If b 4ac<, the roots are imaginary because we cannot take the square root of a b ± negative number negative number. Looking at the quadratic formula: x = a Module A6 More on Quadratics - 3
Example 3 Determine the discriminant for the following quadratic equation. Use this to solve the equation. 3x x + 5 = Here a = 3, b =, c = 5 b 4ac= = 56 ( ) 4(3)(5) Since the discriminant is less than zero, the roots are imaginary. Don t try to solve by factoring. Instead go directly to the quadratic formula. ± x = 4 b b ac a ± 56 = 6 + 7.48i 7.48i = and 6 6 =.33 + 1.5 i and.33 1.5i Check your answer. SUMMARY The discriminant is a number that will assist you in solving a quadratic equation: If the discriminant is, then the roots are real and equal. If the discriminant is >, then the roots are real and unequal. If the discriminant is <, then the roots are imaginary. 4 Module A6 More on Quadratics -
Experiential Activity One Determine the discriminant for each of the following quadratic equations. Use this to help you solve the given equations. 1. x + x 3 =. x + x 4 = 3. x 3x + 1 = 4. x 5x + 5 = 5. t + 3t + 1 = 6. z + z = 7. x x = 1 8. x x = 9. 3t + 6t = 1. 3x + 6x + 1 = 11. x x x x 1 = 1 1. + = 6 3 1 3 1 13. x + 5 = 1x 14. y + 3 = 1y 15. y + y + = 16. x x + = 17. x 7x + 4 = 18. 3x 5x = 4 19. t + 1t = 15. t + 7 = 4t 1. 4x 9 =. x 6x = 3. 15 + 4x 3x = 4. 4x 1x = 7 Show Me. 5. 1 1 1 x + x = 6. 1 1 1 x x = 7. 4 3 1 3 1 x x= 8. 3 4 3 1 4 1 x x= 3 9 3 Experiential Activity One Answers 1. 13. 17 3. 5 4. 5 5. 1 6. 17 7. 8 8. 1 9. 1 1. 4 11..7778 1..833 13. 44 14. 8 15. 4 16. 4 17. 17 18. 73 19.. 4 1. 144. 36 3. 1936 4. 56 5..1944 6..1944 7. 3 8. 1.864 Module A6 More on Quadratics - 5
OBJECTIVE TWO When you complete this objective you will be able to Determine the quadratic equation given the roots of the equation. Exploration Activity Example 1 Determine the quadratic equation given the roots as: then x = 6, 1 x = 6, and x = 1 a) Determine factors - move everything to the Left Hand Side (LHS) x + 6 = and x 1 = b) Multiply the factors. ( )( ) x+ 6 x 1 = x x+ 6x 6= c) Simplify x + 5x 6= d) Check by substitutions substitute the roots from above in and solve. for x = 6 ( ) ( ) 6 + 5 6 6= 36 3 6 = = for x = 1 Both Checks Work!! () () 1 + 5 1 6= 1+ 5 6= = 6 Module A6 More on Quadratics -
Example Determine the quadratic equation given the roots are: 5 1 x =, 3 Find the quadratic for 5 1 x= and x= 3 a) Determine factors - clear denominators first. 3x = 5 and x = 1 - move everything to the LHS 3x 5 = and x +1 = b) Multiply the factors. ( x )( x ) 3 5 + 1 = 6x + 3x 1x 5= c) Simplify 6x 7x 5= d) Check by substitutions Complete this on your own. Module A6 More on Quadratics - 7
Example 3 Determine the quadratic equation given the roots are: x = 3± 5 Find the quadratic for x= 3+ 5 and x= 3 5 a) Determine factors x = 3+ 5 x 3= 5 x 3 5 = x = 3 5 x 3= 5 x 3+ 5 = b) Multiply the factors. ( x )( x ) 3 5 3+ 5 = x x x x c) Simplify x 3 + 5 3 + 9 3 5 5= 6x+ 4= d) Check by substitutions Complete this on your own. 8 Module A6 More on Quadratics -
Experiential Activity Two Determine the quadratic equation for each of the following pairs of given roots. 1. x = 1, 3. x =, 3 3. x =, 4 4. x = 6, 5. x = 4, 4 6. x =, 5 7. 5 5 x =, 8. x = 1, 9. x = 4, 6 1. x = 1± 7 Show Me. 11. x = 4± 3 1. x = 1± Experiential Activity Two Answers 1. x 4x + 3 =. x + 5x + 6 = 3. x 6x + 8 = 4. x + 4x 1 = 5. x 16 = 6. x 5x = 7. x 6.5 = 8. x 3x + = 9. x + 1x + 4 = 1. x x 6 = 11. x 8x + 13 = 1. x x 1 = Module A6 More on Quadratics - 9
OBJECTIVE THREE When you complete this objective you will be able to Determine the quadratic equation given the sum and product of the roots. Exploration Activity A quick way to determine an equation from the roots is to use the sum and product of the roots. Determining an Equation from the sum and product of the roots Example 1 Given the general quadratic equation: ax + bx + c =. If we know the roots of the equation, coefficient b is the negative of the sum of the roots and coefficient c is the product of the roots. if the roots are r 1 and r then coefficient b is: (r 1 + r ) the neg. of the sum of the roots. coefficient c is: r 1 r the product of the roots. The equation can therefore be determined from the roots using the following: ( ) x r1 + r x+ rr 1 = x or + ( negative of sum of roots) x + product of roots = Construct the quadratic equation which has roots of and 3. Find the negative of the sum of the roots: (r 1 + r ) = (r 1 + r ) = ( + 3) = 5 Therefore coefficient b is 5 Find the product of the roots: r 1 r = r 1 r = 3 = 6 Therefore coefficient c is 6 Substitute into x + bx+ c= x 5x+ 6= 1 Module A6 More on Quadratics -
Example Construct the quadratic equation which has roots of 3 and 4. Find the negative of the sum of the roots: (r 1 + r ) = (r 1 + r ) = ( 3 + 4) = 1 Therefore coefficient b is 1 Find the product of the roots: r 1 r = r 1 r = 3 4 = 1 Therefore coefficient c is 1 Substitute into x + bx+ c= x 1x 1= Example 3 Construct the quadratic equation which has roots of 3 and. Find the negative of the sum of the roots: (r 1 + r ) = (r 1 + r ) 3 + = ( ) = ( 3 4 ) 1 = ( ) = 1 Therefore coefficient b is 1 Find the product of the roots: r 1 r = r 1 r = 3 = 3 Therefore coefficient c is 3 Substitute into x + bx+ c= 1 x + x 3= Clear the fractions by multiplying by the Lowest Common Denominator (LCD) x + x 6= Module A6 More on Quadratics - 11
Experiential Activity Three Find the quadratic equation given the sum and product of the roots. SUM PRODUCT 1. 7 6. 3 1.5 3. 3 5 4. 5/3 5. 1.5 6. m n + n m 1 Find the equation given the roots by first determining the sum and products of the roots. 7. x = 1, 4 8. x = 3, 7 9. x =, 1 3 1. x = + 3, 3 Experiential Activity Three Answers 1. x 7x + 6 =. x + 6x 3 = 3. x + 3x + 5 = 4. 3x 5x = 5. x + 3 = 6. mnx + (m + n )x + mn = Show Me. 7. x 5x+ 4= 8. x 4x 1= 9. 6x x = 1. x 4x+ 1= Practical Application Activity Complete the Quadratics Equations - assignment in TLM. Summary This module presents the student with additional theory on quadratic equations. 1 Module A6 More on Quadratics -