Page 32 - 11-Math-4 Quadratic Equations

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vii) (a - β )2 , (a + β )2 viii) - 1 ,- 1
α3 β3

8. If α , β are the roots of the 5x2 - x - 2 = 0, form the equation whose roots are

3 and 3 .
α β

9. If α , β are the roots of the x2 - 3x + 5 = 0, form the equation whose roots are

1-α and 1- β .
1+α 1+ β

4.10 Nature of the roots of a quadratic equation

We know that the roots of the quadratic equation ax2 + bx + c = 0 are given by the

quadratic formula as: x = -b ± b2 - 4ac
2a

We see that there are two possible values for x, as discriminated by the part of the
formula ± b2 - 4ac .

The nature of the roots of an equation depends on the value of the expression b2 - 4ac,
which is called its Discriminant.

Case 1: If b2 - 4ac = 0 then the roots will be - b and - b So, the
roots are real and repeated equal. 2a 2a

Case 2: If b2 - 4ac < 0 then b2 - 4ac will be imaginary
So, the roots are complex / imaginary and distinct / unequal

Case 3: If b2 - 4ac > 0 then b2 - 4ac will be real.
So, the roots are real and distinct / unequal.

However, If b2 - 4ac is a perfect square then b2 - 4ac will be rational, and so the
roots are rational, otherwise irrational.

version: 1.1

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