F4 Maths

Find the values of (α - 1)(β - 1).

(α - 1)(β - 1) = αβ - (α+β) + 1 = - 3/4 - (-6/4) + 1 = 7/4
Form a quadratic equation in x with roots α/α – 1 and β/β –1.
Method 1 :
α/(α - 1) + β/(β - 1)

　α(β - 1) + β(α - 1)
= ────────────
　　(α - 1)(β - 1)

　2αβ - (α+β)
= ────────
　(α - 1)(β - 1)

　2(-3/4) - (-6/4)
= ─────────
　 　　 7/4
= 0
;
α/(α - 1) × β/(β - 1)

= αβ / ( (α - 1)(β - 1) )

= (- 3/4) / (7/4)

= - 3/7

∴ The required equation is x² - 0x + (-3/7) = 0
7x² - 3 = 0


Method 2 :
Let x = α/(α - 1)
xα - x = α
α(x - 1) = x
α = x/(x - 1) , similarly β = x/(x - 1).
∵ α and β are the roots of the equation 4x² + 6x - 3 = 0
∴ The required equation is 4 x²/(x - 1)² + 6 x/(x - 1) - 3 = 0
4x² + 6x(x - 1) - 3(x - 1)² = 0
4x² + 6x² - 6x - (3x² - 6x + 3) = 0
7x² - 3 = 0