TMUA Algebraic Techniques & Manipulation 3 — Questions and Answers
Question 1: Factorise completely: 2x³ - 8x
- 2x(x-2)(x+2) (Correct answer)
- 2x(x²-4)
- x(2x-4)(x+2)
- 2(x³-4x)
Correct 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).
Question 2: Solve the simultaneous equations: x + 2y = 7 and x² + y² = 25.
- (3,2) and (7,0)
- (5,1) and (3,2)
- (3,2) and (-7,7) (Correct answer)
- (0,5) and (7,0)
Correct 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.
Question 3: Write 3x² + 12x + 7 in completed square form.
- 3(x+2)² - 5 (Correct answer)
- 3(x+2)² + 7
- (3x+6)² - 5
- 3(x+4)² - 5
Correct 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.
Question 4: Simplify: (x^(1/2) + x^(-1/2))²
- x + 2 + x⁻¹ (Correct answer)
- x + x⁻¹
- x² + 2 + x⁻²
- 2x + 2
Correct answer: x + 2 + x⁻¹
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.
- x² - 8x - 12 = 0 (Correct answer)
- x² - 4x - 12 = 0
- x² - 8x - 3 = 0
- x² + 8x - 12 = 0
Correct answer: x² - 8x - 12 = 0
New sum = 2(α+β) = 8, new product = 4αβ = -12, giving x²-8x-12=0.
Question 6: Solve: x⁴ - 13x² + 36 = 0
- x = ±2, ±3 (Correct answer)
- x = ±4, ±9
- x = ±1, ±6
- x = ±2, ±6
Correct 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.
Question 7: Simplify: (a - b)/(a² - b²) × (a + b)/(a - b)
- 1/(a-b) (Correct answer)
- 1/(a+b)
- (a+b)/(a-b)²
- 1
Correct 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).
TMUA Algebraic Techniques & Manipulation 3
Factorise completely: 2x³ - 8x