Completing the Square

Completing the square rewrites $ax^2 + bx + c$ in the form $a(x + p)^2 + q$. It’s the most-requested “express in the form” task in 0606, and the completed form is a master key that opens four other question types.

The routine, $a = 1$

$x^2 - 10x + 18 = (x - 5)^2 - 25 + 18 =$ $(x - 5)^2 - 7$

Half the $x$-coefficient goes in the bracket; subtract its square; tidy. Two visible steps, both creditable.

The routine, $a \ne 1$, factor out first

The slip zone. Factor $a$ from the $x$-terms before halving:

$3x^2 + 12x - 5$ $= 3(x^2 + 4x) - 5$ $= 3[(x + 2)^2 - 4] - 5$ $= 3(x + 2)^2 - 12 - 5$ = $3(x + 2)^2 - 17$

The $-4$ inside the bracket gets multiplied by $3$ on the way out, forgetting that multiplication is the classic error, and it’s exactly what the line-by-line layout catches. For negative $a$ (e.g. $7 + 8x - 2x^2$), factor out $-2$ from the $x$-terms and track signs doubly carefully; the result has a maximum.

What the form is for

With $f(x) = a(x + p)^2 + q$ in hand:

1. Vertex: $(-p, q)$; max/min value $q$ at $x = -p$, see maximum/minimum & vertex
2. Range: $f(x) \ge q$ ($a > 0$) or $\le q$ ($a < 0$), see range of a quadratic
3. Solving exactly: $a(x + p)^2 + q = 0 \to x = -p \pm \sqrt{-q/a}$, the surd-friendly route on Paper 1
4. Root counting: if $q > 0$ and $a > 0$ the expression never reaches zero, an elegant “show it has no real roots” argument

“Express in the form” is a format command: the accuracy marks attach to that exact shape, with $a$, $p$, $q$ identified. Check your answer in seconds by expanding back.

Common mistakes

• Halving $b$ before factoring out $a$
• The subtracted square left unmultiplied by $a$
• Sign loss with negative $a$
• Stopping at the form when the question continues (“hence state the minimum…”), read the whole question

Full topic context: Quadratic Functions notes.