TMUA Algebraic Techniques & Manipulation 3

Question 1

Factorise completely: 2x³ - 8x

Accepted Answer

2x(x-2)(x+2)

Answer

Factor out 2x to get 2x(x²-4), then factor the difference of squares to get 2x(x-2)(x+2).

Question 2

Solve the simultaneous equations: x + 2y = 7 and x² + y² = 25.

Accepted Answer

(3,2) and (-7,7)

Answer

Substituting x=7-2y into x²+y²=25 gives 5y²-28y+24=0, yielding y=2,x=3 and y=7,x=-7.

Question 3

Write 3x² + 12x + 7 in completed square form.

Accepted Answer

3(x+2)² - 5

Answer

Factor out 3: 3(x²+4x)+7 = 3[(x+2)²-4]+7 = 3(x+2)²-12+7 = 3(x+2)²-5.

Question 4

Simplify: (x^(1/2) + x^(-1/2))²

Accepted Answer

x + 2 + x⁻¹

Answer

Expanding: x + 2·x^(1/2)·x^(-1/2) + x^(-1) = x + 2 + x⁻¹.

Question 5

Given that α + β = 4 and αβ = -3, form a quadratic with roots 2α and 2β.

Accepted Answer

x² - 8x - 12 = 0

Answer

New sum = 2(α+β) = 8, new product = 4αβ = -12, giving x²-8x-12=0.

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