Completing the Square Worksheet

This completing the square worksheet helps students practice turning quadratic expressions into perfect-square trinomials and solving from there. The printable set includes 12 randomized questions and an answer key on page 2.

What is Completing the Square?

Completing the square is a method for rewriting a quadratic expression so that it becomes a perfect square plus or minus a constant. This makes some quadratic equations easier to solve and helps connect equations to vertex form.

How to Complete the Square

1. Move the constant term to the other side.
2. Take half of the coefficient of $x$.
3. Square that number.
4. Add it to both sides of the equation.
5. Rewrite the left side as a square and solve.

Example

Solve $x^2 + 6x - 7 = 0$

Move the constant:

$x^2 + 6x = 7$

Take half of $6$, which is $3$, and square it to get $9$.

Add $9$ to both sides:

$x^2 + 6x + 9 = 16$

Rewrite:

$(x + 3)^2 = 16$

So $x + 3 = \pm 4$, which gives $x = 1$ or $x = -7$.

FAQ

Q: Why do we add the same number to both sides?
A: Adding the same number to both sides keeps the equation balanced.

Q: When is completing the square useful?
A: It is especially helpful when factoring is difficult and when rewriting a quadratic into vertex form.