Construct a quadratic equation whose roots are 3 and - 8 .

Given that the roots of the equation are 3 , - 8 The quadratic equation is given by x 2 - ( α + β ) x + α β = 0 , where α , β are roots of equation. So, if we construct a quadratic equation by given roots, we get x 2 - ( 3 + - 8 ) x + 3 ( - 8 ) = 0 ⇒ x 2 + 5 x - 24 = 0 Hence, the quadratic equation with roots 3 and - 8 is x 2 + 5 x - 24 = 0

Find the nature of the roots of the quadratic equation 4 x 2 - x + 1 = 0 .

Given, 4 x 2 - x + 1 = 0 Comparing the given quadratic equation with the general form of a quadratic equation a x 2 + b x + c = 0 , we get, a = 4 , b = - 1 and c = 1 Discriminant, D = b 2 - 4 a c = - 1 2 - 4 × 4 × 1 = 1 - 16 = - 15 Here, D < 0 If the discriminant of a quadratic equation is less than 0 , then the given equation has no real roots. Hence, the quadratic equation 4 x 2 - x + 1 = 0 does not have real roots.

Solve the quadratic equation: x 2 - 9 x = 0

Given: x 2 - 9 x = 0 On taking out x common, x x - 9 = 0 ∴ x = 0 and x - 9 = 0 x = 0 and x = 9 Hence, the roots of given quadratic equation are x = 0 and x = 9 .