How to Complete the Square: Step-by-Step Guide With Examples for 2026

Completing the square is the most powerful, most under-used quadratic technique in high school. It is the only method that solves quadratics, derives the quadratic formula, converts standard form to vertex form, and rewrites conic sections. Students often dread it because the steps look longer than factoring, but with the right recipe it takes about 90 seconds per problem.

This guide gives you a clean recipe, walks through every variation (leading coefficient one, leading coefficient other than one, conics), and ends with the five mistakes that cost most students points.

What Completing the Square Actually Does

Completing the square rewrites a quadratic expression as a perfect square plus a constant. A perfect square trinomial looks like:

(x + p)² = x² + 2px + p².

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If the middle coefficient is 2p, the constant that finishes the square is p². You take half the middle coefficient and square it.

The Recipe: Five Steps

For ax² + bx + c with a = 1:

1. Move the constant to the other side: x² + bx = −c.
2. Take half of b. Call it p.
3. Square p and add p² to both sides: x² + bx + p² = −c + p².
4. Factor the left as (x + p)².
5. Solve by taking the square root and isolating x.

Five steps. Same recipe every time.

Example 1: Leading Coefficient 1

Solve x² + 6x − 7 = 0.

Step 1. Move −7: x² + 6x = 7.
Step 2. Half of 6 is 3.
Step 3. 3² = 9. Add to both sides: x² + 6x + 9 = 16.
Step 4. Factor: (x + 3)² = 16.
Step 5. Square root: x + 3 = ±4. So x = 1 or x = −7.

Always include the ± when you take a square root.

Example 2: Negative Middle Coefficient

Solve x² − 10x + 21 = 0.

Step 1. Move 21: x² − 10x = −21.
Step 2. Half of −10 is −5.
Step 3. (−5)² = 25. Add: x² − 10x + 25 = 4.
Step 4. Factor: (x − 5)² = 4.
Step 5. Square root: x − 5 = ±2. So x = 7 or x = 3.

The sign of p matches the sign of b/2. Half of −10 is −5, so the factor is (x − 5), not (x + 5).

Example 3: Leading Coefficient Not 1

Solve 2x² − 8x + 6 = 0.

Step 0 (new step). Divide the entire equation by the leading coefficient.
– Divide by 2: x² − 4x + 3 = 0.

Now run the standard recipe.
Step 1. x² − 4x = −3.
Step 2. Half of −4 is −2.
Step 3. (−2)² = 4. Add: x² − 4x + 4 = 1.
Step 4. (x − 2)² = 1.
Step 5. x − 2 = ±1. So x = 3 or x = 1.

If you cannot cleanly divide, factor the leading coefficient out of just the x terms and leave the constant alone. We will show that in the vertex-form example below.

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Example 4: Irrational Solutions

Solve x² + 4x − 1 = 0.

Step 1. x² + 4x = 1.
Step 2. Half of 4 is 2.
Step 3. 2² = 4. Add: x² + 4x + 4 = 5.
Step 4. (x + 2)² = 5.
Step 5. x + 2 = ±√5. So x = −2 ± √5.

Irrational answers are normal for completing the square. Leave them in exact form unless the problem asks for a decimal.

Converting Standard Form to Vertex Form

Vertex form of a quadratic is y = a(x − h)² + k, where (h, k) is the vertex of the parabola.

Convert y = x² − 8x + 11 to vertex form.

1. Group the x terms: y = (x² − 8x) + 11.
2. Half of −8 is −4; square it: 16.
3. Add and subtract 16 inside: y = (x² − 8x + 16) − 16 + 11.
4. Factor: y = (x − 4)² − 5.

Vertex: (4, −5).

Vertex Form With Leading Coefficient Not 1

Convert y = 2x² + 12x + 7 to vertex form.

1. Factor 2 out of the x terms only: y = 2(x² + 6x) + 7.
2. Half of 6 is 3; square it: 9.
3. Add 9 inside and subtract 2 · 9 = 18 outside: y = 2(x² + 6x + 9) + 7 − 18.
4. Factor and combine: y = 2(x + 3)² − 11.

Vertex: (−3, −11).

Watch the bookkeeping: whatever you add inside the parentheses must be multiplied by the leading coefficient when you subtract it outside.

Completing the Square for Conic Sections

Completing the square is how you convert a general conic equation to standard form.

Convert x² + y² − 4x + 6y − 12 = 0 to standard form.

1. Group: (x² − 4x) + (y² + 6y) = 12.
2. Complete each square. Half of −4 is −2, squared is 4. Half of 6 is 3, squared is 9.
3. Add to both sides: (x² − 4x + 4) + (y² + 6y + 9) = 12 + 4 + 9 = 25.
4. Factor: (x − 2)² + (y + 3)² = 25.

This is a circle with center (2, −3) and radius 5.

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The same approach works for ellipses, hyperbolas, and parabolas, just with the right coefficients.

When to Choose Completing the Square Over Factoring

Deriving the Quadratic Formula

The quadratic formula comes from completing the square on the general quadratic ax² + bx + c = 0. The key steps:

1. Divide by a: x² + (b/a)x + (c/a) = 0.
2. Move c/a: x² + (b/a)x = −c/a.
3. Add (b/(2a))² to both sides.
4. Factor and simplify the right side under a common denominator.
5. Square root both sides and isolate x.

The result: x = (−b ± √(b² − 4ac)) / (2a).

If your teacher ever asks how the formula came from, this is the answer.

Common Completing-the-Square Mistakes

1. Forgetting to add the same constant to both sides. Adding p² to the left without also adding to the right unbalances the equation.
2. Skipping the divide step when a ≠ 1. You must divide first or factor out a, depending on whether you are solving or converting.
3. Dropping the ± after the square root. You will lose half the solutions every time.
4. Sign error inside (x ± p)². Half of a negative b is a negative p, so the factor is (x − p).
5. Mishandling the outside constant when factoring out a. Multiplying through is the most common spot for sign errors in vertex form conversions.

A Memorable Worked Pattern

Almost every completing-the-square problem boils down to: half, square, add, factor, root.

• Half the linear coefficient.
• Square it.
• Add to both sides.
• Factor as a perfect square.
• Take the root and isolate.

Five words. Tape them on the page during practice and the recipe stays in your head.

Frequently Asked Questions

Why bother with completing the square if I can just use the quadratic formula?
Three reasons. First, vertex form requires it. Second, conic standard forms require it. Third, calculus and physics use the technique constantly to simplify integrals and identify maxima.

Is the quadratic formula faster than completing the square?
For numeric answers, usually yes. For vertex form or conics, completing the square is faster and more informative.

Does completing the square always work?
Yes, for every quadratic equation, whether the roots are real or complex.

What if the quadratic has complex roots?
Take the square root of a negative on the right-hand side. The ± becomes ± an imaginary number. Same recipe, complex answer.

Should I leave answers in exact form or as decimals?
Exact form (√5, fractions) unless the problem specifically asks for a decimal approximation.

Closing Thought

Completing the square is five steps and one rule (whatever you add to one side, add to the other). Master the recipe for leading coefficient one, learn the small detour for leading coefficient other than one, and you unlock vertex form, conic standard forms, and the quadratic formula itself. It is the most valuable 90 seconds of quadratic work in high school math.

For more practice, browse our Algebra 1 worksheets and our full Math Topics library. When you are ready for a structured workbook, our Algebra 2 collection covers completing the square and every related topic above.

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