TMUA Algebraic Techniques & Manipulation 2

Question 1

Simplify: (x² - 9) / (x² - x - 6)

Accepted Answer

(x+3)/(x+2)

Answer

Factor numerator as (x-3)(x+3) and denominator as (x-3)(x+2), then cancel (x-3) to get (x+3)/(x+2).

Question 2

If p and q are roots of x² - 5x + 3 = 0, find p² + q².

Accepted Answer

19

Answer

Using p+q=5 and pq=3, p²+q² = (p+q)² - 2pq = 25 - 6 = 19.

Question 3

Solve for x: |2x - 3| = 7

Accepted Answer

x = 5 or x = -2

Answer

Setting 2x-3=7 gives x=5; setting 2x-3=-7 gives x=-2.

Question 4

Expand and simplify: (2x + 3)² - (2x - 3)²

Accepted Answer

24x

Answer

Using difference of squares: [(2x+3)+(2x-3)][(2x+3)-(2x-3)] = (4x)(6) = 24x.

Question 5

Find all values of k such that x² + kx + 9 = 0 has exactly one real solution.

Accepted Answer

k = ±6

Answer

For one real solution, discriminant = 0: k² - 36 = 0, so k = ±6.

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