Completing the Square | Moosa Academy

A quadratic equation is hard to solve because the unknown appears both squared and on its own. Completing the square removes that difficulty by rewriting the equation as a single perfect square, after which taking a square root finishes the job.

Concept The idea

fills all of the square except one corner.

That corner has area

Add it and the figure becomes the complete square

The name is literal: an incomplete square is being completed. Recall the identity — this method runs it backwards.

Note The three steps

Make the coefficient of equal to 1, dividing the whole equation if necessary.
Take the coefficient of , halve it, square it, and add that to both sides.
Write the left side as a squared bracket and take the square root of both sides, remembering .

Adding to only one side is the most common mistake, and it changes the equation.

Example A leading coefficient of 1

Solve .

The coefficient of

is already 1.

Half of 6 is 3, and

. Add 9 to both sides:

Move the constant across:

⟹ x = −1 or x = −5

Example A leading coefficient other than 1

Solve .

Divide everything by 2:

Half of 4 is 2, and

. Add 4 to both sides:

⟹ x = 1 or x = −5

Dividing first is essential — the halve-and-square rule assumes the leading coefficient is 1.

Example When there is no real solution

Solve .

Half of 4 is 2, and

. Add 4 to both sides:

No real number squares to a negative result.

⟹ no real solutions

The method reports the failure clearly, and the negative right-hand side is exactly what a negative discriminant means.

Example A rectangle

A rectangle is 3 units longer than it is wide and has an area of 40 square units. Find its dimensions.

Let the width be

, so the length is

, that is

Half of 3 is 1.5, and

. Add it to both sides:

, so

A width cannot be negative, so the second root is rejected.

⟹ width 5, length 8

Summary

Make the coefficient of x² equal to 1 before doing anything else.
Halve the coefficient of x, square it, and add the result to both sides.
Write the left side as (x + m)² and take the square root, keeping the ± sign.
A negative right-hand side means there are no real solutions.