Choosing the Best Method

≈ 25 min

Four methods now live in your toolkit. Choosing well saves minutes in an exam:

Equation looks like	Best method
$x^2 = k$ or $(x+p)^2 = k$	Square roots — instant
Factorises easily (nice integers)	Factorising — fast and clean
Stubborn integers, or "give exact/surd form"	Completing the square or formula
Anything, anytime	Formula — the universal fallback

Drill. $x^2 = 50$ → square roots. $x^2 + 5x + 6 = 0$ → factorises ( $2, 3$ ). $x^2 + 4x + 1 = 0$ → product 1, sum 4? No integer pair — formula or complete the square.

A 30-second factor-hunt is worth it: if nothing bites, switch to the formula without guilt.

Most efficient method for $x^2 = 50$ ?

Quadratic formula

Factorising

Square roots directly

Completing the square

Most efficient method for $x^2 + 5x + 6 = 0$ ?

Factorising — the factors are small integers

Square roots directly

Completing the square

Graphing precisely

For $x^2 + 4x + 1 = 0$ , why is factorising a poor choice?

The equation is not quadratic

No pair of integers multiplies to 1 and adds to 4

Factorising never solves equations

Because $a = 1$

Solve using any method: $x^2 - 2x - 8 = 0$ . Answer in the form x=… or x=…