Algebraic Techniques & Manipulation Flashcards

Factorise completely: 2x³ - 8x

Answer: 2x(x-2)(x+2)

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

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

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

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

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

Answer: 3(x+2)² - 5

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

Simplify: (x^(1/2) + x^(-1/2))²

Answer: x + 2 + x⁻¹

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

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

Answer: x² - 8x - 12 = 0

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

Solve: x⁴ - 13x² + 36 = 0

Answer: x = ±2, ±3

Substituting u=x² gives u²-13u+36=0, so (u-4)(u-9)=0, thus x²=4 or x²=9, giving x=±2,±3.

Simplify: (a - b)/(a² - b²) × (a + b)/(a - b)

Answer: 1/(a-b)

The product equals (a-b)(a+b)/[(a-b)²(a+b)] = (a+b)/[(a-b)(a+b)] wait — simplifying gives 1/(a-b).