Quadratic equations: three methods, and when to use each

A quadratic is any equation that rearranges to

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.

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:

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

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:

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.