What you need to know
Completing the square rewrites a quadratic so its squared structure is visible. It is especially useful for finding a turning point, solving equations that do not factorise neatly and analysing quadratic inequalities.
- For $x^2+bx$, add and subtract $(b/2)^2$.
- If the coefficient of $x^2$ is not 1, factor it from the quadratic terms first.
- Use the completed-square form to identify the minimum or maximum value.
- When solving inequalities, interpret the squared expression carefully rather than taking one square root only.
Key mathematics
Monic quadratic
x^2+bx+c=\left(x+\frac b2\right)^2+c-\left(\frac b2\right)^2
Half the coefficient of x, then square it.General quadratic
ax^2+bx+c=a\left(x+\frac{b}{2a}\right)^2+c-\frac{b^2}{4a}
Factor a from the x-squared and x terms first.Worked example
Complete the square for $2x^2+12x+7$: $$2(x^2+6x)+7=2\left((x+3)^2-9\right)+7=2(x+3)^2-11.$$ The minimum value is $-11$ and occurs when $x=-3$.
EXAM TECHNIQUE
Exam tips
- Write the intermediate correction term; it makes sign errors easier to catch.
- Expand your final completed-square form once as a quick verification when accuracy matters.
- For a quadratic inequality, use the graph/interval interpretation after finding boundary values.
WATCH OUT
Common mistakes
- Forgetting to multiply the correction term by the factor outside the bracket.
- Using $b^2$ instead of $(b/2)^2$ for a monic quadratic.
- Concluding $x>a$ only from $(x-h)^2>a^2$; there are usually two intervals.
Frequently asked questions
Why do we add and subtract the same number?
It creates a perfect square without changing the value of the original expression.
What if the coefficient of x squared is not 1?
Factor that coefficient from the quadratic and linear terms before completing the square inside the bracket.
How can I check my answer?
Expand the completed-square form and compare the coefficients with the original quadratic.