Nature of the roots of a quadratic equation

powerpoint presentation

1. Nature of the Roots of a Quadratic Equation Ms. G. Martin

2. The number represented by b 2 – 4ac = 0 is called the discriminant of the quadratic equation. How do we determine the nature of the roots of a quadratic equation without actually solving the equation? The nature of the roots can be determined by finding the value of the discriminant.

3. Case 1. If b 2 – 4ac = 0, then the roots of ax 2 + bx + c = 0 are real and equal . Example 1. Determine the nature of the roots of x 2 – 6x + 9 = 0 a = 1, b = -6, c = 9 b 2 – 4ac = ( -6) 2 – 4(1)(9) = 36 – 36 = 0 Since b 2 – 4ac = 0, then x 2 – 6x + 9 = 0 has two real roots which are equal.

4. Verifying that then x 2 – 6x + 9 = 0 has two equal real roots, By factoring, we have then x 2 – 6x + 9 = Factor: (x – 3)( x – 3) = 0 roots : x = 3 ; x = 3

5. Case 2: If b 2 – 4ac > 0, then the roots of ax 2 + bx + c = 0 are real and unequal . Example 1. Determine the nature of the roots of x 2 – 8x + 15 = 0. a = 1, b = -8, c = 15 b 2 – 4ac = (-8) 2 – 4(1)(15) = 64 - 60 = 4 Since b 2 – 4ac > 0, then x 2 – 8x + 15 = 0 has two real roots which are unequal.

6. Verify that x 2 – 8x + 15 = 0 has two unequal real roots: By factoring we have, (x – 3)(x - 5) = 0 X = 3 and x = 5

7. Case 3: If b 2 – 4ac < 0, then the roots of ax 2 + bx + c = 0 are imaginary and unequal . Example 3. Determine the nature of the roots of x 2 – 2x + 2 = 0. a = 1, b = -2, c = 2 b 2 – 4ac = (-2) 2 – 4(1)(2) = 4 – 8 = -4 Since b 2 – 4ac < 0, then x 2 – 2x + 2 = 0 has two unequal imaginary roots.

8. By quadratic formula we have, x 2 – 2x + 2 = 0, a = 1, b = -2 c = 2 x = x = x = x = and x = x = 1 + 1i and x = 1 - 1i x = 1 + i and x = 1 - i

9. THANK YOU ! SMSDGRADE9-MT . OLYMPUS@ODL2020