Completing the Square

Completing the square turns a quadratic expression into a form whose geometry and algebra are easier to read.

The perfect-square pattern

Expanding a squared binomial gives

Therefore an expression $x^2+bx$ becomes part of a perfect square when we add

Half the coefficient of $x$, then square it. This single pattern drives the method.

Rewriting a monic quadratic

To rewrite $x^2+bx+c$:

1. group $x^2+bx$;
2. add and subtract $(b/2)^2$ so the value does not change;
3. factor the resulting perfect-square trinomial;
4. combine the constants.

For example,

The expression is unchanged, but the vertex structure is now visible.

Worked example: convert to vertex form

When the leading coefficient is not one

If $a\ne1$, first factor $a$ from the quadratic and linear terms only.

Rewrite $2x^2+12x+7$:

Complete the square inside the brackets:

Forgetting that the added square is inside a factor of $2$ changes the expression.

Solving a quadratic by completing the square

The method can isolate a square and then use square roots. Solve

Move the constant and complete the square:

Take both square-root branches:

The $\pm$ is necessary because both a positive and negative number can have the same square.

Why it reveals the vertex

In vertex form $y=a(x-h)^2+k$, the square is never negative. If $a>0$, its smallest value occurs when $x=h$, so the minimum point is $(h,k)$. If $a<0$, the vertex is a maximum.

Completing the square therefore changes an expanded formula into a form that directly displays the turning point, axis of symmetry, opening direction, and vertical stretch.

Connection to the quadratic formula

The quadratic formula can be derived by completing the square on

After dividing by $a$, moving the constant, and completing the square, solving for $x$ produces

This explains why the discriminant appears: it is the expression left under the square root.

Choosing among quadratic methods

Factoring is fastest when factors are obvious. The quadratic formula works uniformly. Completing the square is especially useful when vertex form or geometric features are required, or when a quadratic lacks easy integer factors.

All three methods agree because they transform the same equation without changing its solutions.

Common mistakes

Adding a number without subtracting it. Rewriting an expression must preserve its value.

Using half of $b$ instead of its square. Add $(b/2)^2$.

Factoring $a$ from the constant too. Initially factor it only from the $x^2$ and $x$ terms unless you deliberately factor the entire expression.

Forgetting the outside multiplier. A square added inside brackets is also multiplied by $a$.

Taking only one square root. Solving $(x-h)^2=q$ requires $x-h=\pm\sqrt q$ when $q>0$.

Quick self-check

• Is the coefficient of $x^2$ inside the working brackets equal to $1$?
• Did I half the linear coefficient and square the result?
• Did I preserve equality by balancing what was added?
• Does expanding the final form reproduce the original?
• If solving, did I include both square-root branches?

Related topics

• Vertex Form
• Quadratic Function
• Quadratic Formula
• Discriminant