Maths Qn On Quadratic Equations and Inequalities?

Question

The roots fo the quadratic equation 16x^2 + 7x + 4 = 0 are α^2 and β^2 . Find

(i) a quadratic equation whose roots are 1/α^2 and 1/β^2 ,

(ii) two distinct quadratic equations whose roots are α and β .

Answer 1

α² ּ β² = 1/4
α² + β² = -7/16

(i) (1/α²) ּ (1/β²) = 4
(1/α²) + (1/β²) = (α² + β²)/(α² ּ β²) = -7/4
EQUATION: x² + (7/4)x + 4 = 0 or x² + 7x + 16 = 0.

(ii) α ּ β = ± ½
Suppose α ּ β = ½
(α + β)² = α² + 2αβ + β² = -7/16 + 1 = 9/16
α + β = ±¾.

Suppose α ּ β = -½
(α + β)² = α² + 2αβ + β² = -7/16 - 1 = -23/16 (Not possible.)

EQUATION 1: x² - (3/4)x + ½ = 0 or 4x² - 3x + 2 = 0
EQUATION 2: x² + (3/4)x + ½ = 0 or 4x² + 3x + 2 = 0

d:

Answer 2

First you need to solve the equation for roots. This doesn't look factorable or a special from, so you will probably have to go the quad formula route and get the roots. Say they are A and B (they may be imaginary). Then for (i), you take the product of (x-1/A)(x-1/B) and express as a quadratic. You would do a similar thing for (ii), but I am not quite sure of the two distinct equations.