F.4 Maths

1.

(a)

[Recall: For a quadratic equation ax² + bx + c = 0, sum of roots = -b/a, product of roots = c/a]
∴ α + β = -(-6)/4 = 3/2 - (1)
αβ = 7/4 - (2)

(1)²:

(α + β)² = (3/2)²
α² + 2αβ + β² = 9/4

Sub. (2) into the equation above:

α² + 2(7/4) + β² = 9/4
α² + β² = 9/4 - 14/4 = 5/4

∴ α² + β² = 5/4.

(b)

As α² and β² are the roots of the required equation, the required equation:
(x - α²)(x - β²) = 0
⇒ x² - α²x - β²x + α²β² = 0
⇒ x² - (α² + β²)x + (αβ)² = 0
⇒ x² - (5/4)(x) + (7/4)² = 0 (from (2) and the answer in part (a))
⇒ 16x² - 20x + 49 = 0

∴ The required equation: 16x² - 20x + 49 = 0