Lesson 1.4 - The Nature of Roots of Quadratic Equations.pptx

Characteristic of Roots Of Quadratic Equations

1. Lesson 1.4 The Nature of Roots of Quadratic Equations

2. Nature of the Roots If a , b , and c are real numbers, and D = b 2 – 4 ac , then the roots of the quadratic equation ax 2 + bx + c = 0 are: real and unequal , if D > 0 ; real and equal , if D = 0 ; or Imaginary and unequal , if D < 0 . Moreover, the roots are: Rational , if D is a perfect square ; Irrational , if D is not a perfect square .

3. Example 1 : Determine the nature of the roots of each equation. a. x 2 – 6 x + 9 = 0 Solution : x 2 – 6 x + 9 = 0 a = 1 ; b = – 6 ; c = 9 D = b 2 – 4 ac → D= (– 6) 2 – 4(1)(9) = 36 – 36 = Since D = 0 , the roots of x 2 – 6 x + 9 = 0 are real , equal and rational .

4. b. x 2 + 6 x + 5 = 0 Solution : x 2 + 6 x + 5 = 0 a = 1 ; b = 6 ; c = 5 D = b 2 – 4 ac → D = (6) 2 – 4(1)(5) = 36 – 20 = 16 Since D = 16 , which is a perfect square, the roots of x 2 + 6 x + 5 = 0 are real , unequal and rational . c. 2 x 2 – 4 x + 5 = 0 Solution : 2 x 2 – 4 x + 5 = 0 a = 2 ; b = – 4 ; c = 5 D = b 2 – 4 ac → D= (– 4) 2 – 4(2)(5) = 16 – 40 = – 24 Since D = – 24 , the roots of 2 x 2 – 4 x + 5 = 0 are imaginary and unequal .

5. d. 5 x 2 – x – 2 = 0 Solution : 5 x 2 – x – 2 = 0 a = 5 ; b = – 1 ; c = – 2 D = b 2 – 4 ac → D= (– 1) 2 – 4(5)(– 2) = 1 + 40 = 41 Since D = 41 , the roots of 5 x 2 – x – 2 = 0 are real , unequal and irrational .

6. Example 2 : Determine the value of k so that the indicated condition in each equation will be satisfied. a. 3 x 2 + 4 x + k = 0 ; Roots are real and unequal Solution : If the roots of 3 x 2 + 4 x + k = 0 are real , unequal , then D > 0 . a = 3 ; b = 4 ; c = k D = b 2 – 4 ac (4) 2 – 4(3)( k ) > 0 16 – 12 k > 0 – 12 k > – 16

7. b. x 2 + k x – 3 x + k = 0 ; Roots are real and equal Solution : If the roots of x 2 + kx – 3 x + k = 0 [equivalently, x 2 + ( k – 3) x + k = 0 ] are real , equal , then D = . a = 1 ; b = k – 3 ; c = k D = b 2 – 4 ac ( k – 3) 2 – 4(1)( k ) = 0 k 2 – 6 k + 9 – 4 k = 0 k 2 – 10 k + 9 = 0 ( k – 9)( k – 1) = 0 k – 9 = 0 or k – 1 = 0 k = 9 or k = 1

8. If the roots of ax 2 + bx + c = 0 are and , then the sum of the roots is

9. and the product of the roots is

10. Sum and Product of Roots of a Quadratic Equation The sum of the roots of ax 2 + bx + c = 0 is The product of the roots of ax 2 + bx + c = 0 is

11. Example 3 : Find the sum and product of the roots of the following equations. a. x 2 + 2 x – 8 = 0 Solution : In x 2 + 2 x – 8 = 0, a = 1 , b = 2 , c = – 8 The sum of the roots is The product of the roots is

12. Note that the roots of x 2 + 2 x – 8 = 0 determined by factoring ( x + 4) ( x – 2) = 0 x 1 = – 4 or x 2 = 2 Their sum: x 1 + x 2 = – 4 + 2 = – 2 and their product: x 1 ● x 2 = – 4 ● 2 = – 8 Solution :

13. b. 2 x 2 – 7 x – 4 = 0, a = 2, b = – 7, c = – 4 Solution : The sum of the roots is The product of the roots is

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