Completing the Square

Some quadratics do not factor with integers, and The Quadratic Formula can feel like a black box. Completing the square is the method underneath both. It reshapes into a perfect square plus a leftover constant, and once you have a perfect square, the square-root property finishes the job.

The one move to learn is this: to make into a perfect square, add . Everything else in the procedure is bookkeeping around that step.

Why that specific number? A perfect-square trinomial always looks like . Compare that to : the middle term must match , so , and the constant it needs is . You are simply supplying the constant that a perfect square would have had.

The Completing-the-Square Formula

The added constant, , is the square of half the coefficient of . It is exactly what a perfect-square trinomial needs, because — the constant is always the square of half the middle coefficient.

Procedure: Leading Coefficient of 1

To solve :

1. Move the constant to the right side: .
2. Add to both sides.
3. Write the left side as a squared binomial and simplify the right side.
4. Apply the square-root property (with ).
5. Solve for .

Procedure: Leading Coefficient Not 1

If the coefficient is , divide every term by first, then follow the steps above.

Dividing every term includes the constant: . Some textbooks instead factor out of the first two terms and complete the square inside the parentheses; that works too, but it forces you to multiply the completing constant by when you move it across the equals sign, which is one more place to slip. Dividing first keeps the coefficient at and the arithmetic simplest.

When the Middle Coefficient Is Odd

If is odd, is a fraction, and is a fraction over . This is normal — it is not a sign that you took a wrong turn. For : half of is , squared is , and the binomial is . Keep the right side as a single fraction with denominator so that the square root at the end is clean:

The in the denominator becomes , which is why the whole answer lands over .

Worked Example A: Leading Coefficient 1

Solve .

Step 1 — move the constant:

Step 2 — half of is ; . Add to both sides:

Step 3 — write the left side as a square:

Step 4 — square-root property:

Step 5 — solve:

Answer: or .

Worked Example B: An Odd Middle Coefficient

Solve .

Step 1 — move the constant:

Step 2 — half of is ; its square is . Add to both sides:

Step 3 — write as a square:

Step 4 — square-root property:

Step 5 — solve:

Answer: . An irrational answer is normal for a quadratic that does not factor.

Worked Example C: Leading Coefficient Not 1

Solve .

Step 1 — divide every term by :

Step 2 — half of is ; . Add :

Step 3 — square-root property:

Answer: .

Worked Example D: Vertex Form

Rewrite in vertex form and state the vertex.

Step 1 — group the -terms and complete the square inside, adding and subtracting the same amount:

Step 2 — write the square and combine constants:

Vertex form is ; the vertex is . Here the added is subtracted back out in the same line, because this is an expression, not an equation — you cannot add to just one side.

Worked Example E: A Negative Discriminant

Solve by completing the square.

Step 1 — move the constant:

Step 2 — half of is ; . Add to both sides:

Step 3 — square-root property; the right side is negative, so the root is imaginary:

Step 4 — solve:

Answer: , a complex-conjugate pair — the same result the quadratic formula would give from the discriminant .

Deriving the Quadratic Formula

Completing the square on in general:

That last line is the quadratic formula — one completing-the-square calculation, done once and for all.

Common Mistakes to Avoid

• Adding to only one side of an equation. Both sides get it. (In an expression, you add and immediately subtract it — see Worked Example D.)
• Forgetting to divide by first when the leading coefficient isn’t 1.
• Squaring instead of . You halve first, then square.
• Dropping the at the square-root step.
• Sign error on . For , the binomial is , not .
• Not simplifying the right side before taking the square root — combine the constants first.

Where This Shows Up Later

• The Quadratic Formula is this method, pre-computed.
• Graphing parabolas. Vertex form, produced by completing the square, gives the vertex and the axis of symmetry directly.
• Conic sections. Completing the square on both and puts a circle or ellipse equation into standard form.
• Integration and other later math. Completing the square is a standard preprocessing step for certain integrals and for analyzing quadratic expressions.

Practice Problems

Work each problem yourself before opening the answer.

Problem 1. What constant completes the square for ?

Problem 2. Solve by completing the square.

Problem 3. Solve by completing the square.

Problem 4. Solve by completing the square.

Problem 5. Solve by completing the square.

Problem 6. Solve by completing the square.

Problem 7. Solve by completing the square.

Problem 8. Rewrite in vertex form and give the vertex.

Problem 9. Solve by completing the square.

Problem 10. Solve by completing the square.

Problem 11. What constant completes the square for ?

Problem 12. Solve by completing the square.

Problem 13. Solve by completing the square.

Quick Reference

Once you have , the finish is the square-root property from Quadratic Equations. For a faster route to the same roots, The Quadratic Formula is this method pre-solved. More Algebra lessons are available too.