mathsGrade 9-102 min read

Quadratic equations: three methods, and when to use each

Factorising, the quadratic formula, completing the square — they all solve the same equation. The skill is reading the question and knowing which one is fastest here.

A quadratic is any equation that rearranges to

ax^2 + bx + c = 0.

There are three standard ways to solve it, and they always give the same answer. Choosing well is the difference between a clean two lines and a page of arithmetic.

Method 1 — Factorising (try this first)

If the quadratic factorises nicely, this is the fastest route. Find two numbers that multiply to $ac$ and add to $b$ , split the middle term, and factor.

x^2 + 5x + 6 = 0 \;\Rightarrow\; (x+2)(x+3) = 0 \;\Rightarrow\; x = -2 \text{ or } x = -3.

The trick: a product equals zero only if one of the brackets is zero. So set each bracket to zero separately.

Method 2 — The quadratic formula (always works)

When factorising won't come, the formula never fails:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The part under the root, $b^2 - 4ac$ , is the discriminant, and it tells you about the roots before you finish:

$b^2 - 4ac$	Roots
Positive	Two different real roots
Zero	One repeated root
Negative	No real roots

Method 3 — Completing the square

Slower to solve with, but it's the method that reveals the turning point of the curve, so it earns its keep in graph questions:

x^2 + 6x + 1 = (x+3)^2 - 9 + 1 = (x+3)^2 - 8

The minimum sits at $(-3, -8)$ — you can read the vertex straight off.

Which to pick

Factorises easily? → factorise.
Doesn't, but you need a numerical answer? → quadratic formula.
Asked for the turning point or a min/max? → complete the square.

Rearrange to "= 0" first, every time. All three methods assume the quadratic is set equal to zero. The most common lost mark is solving $x^2 + 5x = 6$ without first moving the $6$ across.

Last revised 28 January 2025.