Algebraic Techniques & Manipulation Flashcards

Rationalise the denominator of: (2 + √3)/(2 - √3)

Answer: 7 + 4√3

Multiply by (2+√3)/(2+√3): numerator = (2+√3)² = 7+4√3; denominator = 4-3 = 1.

Solve: 2^(2x) - 5·2^x + 4 = 0

Answer: x = 0 or x = 2

Let u = 2^x: u²-5u+4=0 gives (u-1)(u-4)=0, so 2^x=1→x=0 or 2^x=4→x=2.

If x + y = 5 and xy = 4, find x³ + y³.

Answer: 65

x³+y³ = (x+y)(x²-xy+y²) = (x+y)[(x+y)²-3xy] = 5[25-12] = 65.

Simplify: log₂(32) - log₂(4) + log₂(2)

Answer: 4

log₂(32)=5, log₂(4)=2, log₂(2)=1; so 5-2+1=4.

Find the value of the expression: (n+1)! / [(n-1)! · n]

Answer: n+1

(n+1)! = (n+1)·n·(n-1)!, so the expression becomes (n+1)·n·(n-1)! / [(n-1)!·n] = n+1.

Solve the inequality: (x-1)(x+3) < 0

Answer: -3 < x < 1

The product is negative when the factors have opposite signs, which occurs for -3 < x < 1.

Given f(x) = x² - 4 and g(x) = x + 2, find f(x)/g(x) simplified, stating any restrictions.

Answer: x - 2, x ≠ -2

f(x)/g(x) = (x-2)(x+2)/(x+2) = x-2, provided x ≠ -2 (since g(-2)=0).