How to Complete the Square (Step-by-Step Guide)

Completing the square is an algebraic technique that transforms a quadratic expression into a perfect square plus a constant. It is one of the most versatile methods in algebra, useful for solving quadratic equations, deriving the quadratic formula, converting parabolas to vertex form, and rewriting circle equations. Once you master this technique, many problems that seem difficult become straightforward.

This guide walks you through the method step by step, with worked examples covering both simple and advanced cases, including quadratics with leading coefficients other than 1.

Why Complete the Square?

You might wonder why we need another method when we already have the quadratic formula and factoring. Completing the square is valuable for several reasons:

• Vertex form: It converts into , which immediately reveals the vertex of the parabola at .
• Deriving the quadratic formula: The quadratic formula itself is derived by completing the square on the general equation .
• Circle equations: It converts the general form of a circle equation into standard form, revealing the centre and radius.
• Integration: In calculus, completing the square is used to evaluate certain integrals involving quadratic expressions.
• Optimisation: It makes finding maximum or minimum values trivial since the vertex form directly shows the extreme point.

The Method: Step by Step

To complete the square on , we want to find a constant that makes this a perfect square trinomial. The key insight is:

So we add and subtract to keep the expression equal. Here is the procedure for a general quadratic :

1. Factor out the leading coefficient (if ): Write .
2. Find the magic number: Take half the coefficient of inside the brackets and square it: .
3. Add and subtract this number inside the brackets.
4. Rewrite the perfect square trinomial as a squared binomial.
5. Simplify the remaining constant terms.

The Completing the Square Formula

For the general quadratic , completing the square gives:

This is the vertex form where:

Worked Examples

Example 1: Simple Case (Leading Coefficient = 1)

Complete the square for .

Step 1: The leading coefficient is already 1, so we focus on the part.

Step 2: Half the coefficient of is . Squaring gives .

Step 3: Add and subtract 9:

Step 4: The first three terms form a perfect square:

The vertex form is , so the vertex of the parabola is at .

Example 2: Solving a Quadratic Equation

Solve by completing the square.

Step 1: Move the constant to the right side:

Step 2: Half of is . Squaring gives .

Step 3: Add 4 to both sides:

Step 4: Take the square root of both sides:

The solutions are and . You can verify by substituting back: and .

Example 3: Leading Coefficient Not Equal to 1

Complete the square for .

Step 1: Factor out the leading coefficient from the first two terms:

Step 2: Inside the brackets, half of 6 is 3, and .

Step 3: Add and subtract 9 inside the brackets:

Step 4: Rewrite the perfect square and distribute:

The vertex form is , with vertex at . Since the leading coefficient is positive, the parabola opens upward and the vertex is a minimum.

Example 4: Solving with a Leading Coefficient

Solve by completing the square.

Step 1: Divide every term by 3 to simplify:

Step 2: Rearrange:

Step 3: Half of is , squared is . Add to both sides:

Step 4: Solve:

Deriving the Quadratic Formula

One of the most elegant applications of completing the square is deriving the quadratic formula. Start with the general equation:

Divide by :

Move the constant:

Add to both sides:

Take the square root:

This is the quadratic formula. Every step in the derivation is an application of completing the square, which is why understanding the technique gives you deeper insight into why the formula works.

Completing the Square for Circle Equations

The general equation of a circle is:

To find the centre and radius, we complete the square separately for and .

Example 5: Converting a Circle Equation

Find the centre and radius of .

Group the and terms:

Complete the square for : half of is , squared is .

Complete the square for : half of is , squared is .

Add these to both sides:

The circle has centre and radius .

Applications Beyond Algebra

Calculus: Integrating Quadratic Expressions

When integrating expressions like , completing the square transforms the denominator into a form suitable for the arctangent formula:

This matches the pattern , making the integral straightforward.

Optimisation Problems

Since vertex form shows the minimum (when ) or maximum (when ) value directly, completing the square is a quick way to optimise quadratic functions without calculus. The extreme value is , occurring at .

Common Mistakes to Avoid

Forgetting to balance both sides: When you add a number to complete the square, you must add the same number to the other side of the equation (or subtract it if you added and subtracted on the same side).

Ignoring the leading coefficient: If , you must factor it out before completing the square. A common error is halving the coefficient of without first dividing by .

Sign errors: Pay careful attention to signs. When the coefficient of is negative, halving it gives a negative number, but squaring that gives a positive result. The binomial will contain the negative sign: , not .

Distributing incorrectly: When you factor out and then subtract the square inside the brackets, remember to multiply by when you bring it back out. For instance, subtracting 9 inside brackets with a factor of 2 means subtracting overall.

Frequently Asked Questions

When should I use completing the square instead of the quadratic formula?

Use completing the square when you need the vertex form of a parabola, when you need to rewrite a circle equation, or when you want to understand the structure of the quadratic. The quadratic formula is faster for simply finding roots, but completing the square gives you more geometric insight. In calculus, completing the square is essential for certain integrals where the quadratic formula does not help.

Does completing the square always work?

Yes. Unlike factoring, which only works neatly when the roots are rational, completing the square works for every quadratic expression. It handles cases with irrational roots, complex roots, and any leading coefficient. This universality is precisely why it can be used to derive the quadratic formula.

How does completing the square relate to the discriminant?

When you complete the square on , you arrive at . The numerator, , is the discriminant. If it is positive, the square root is real and you get two solutions. If it is zero, you get one repeated root. If it is negative, the square root is imaginary and there are no real solutions.

Can you complete the square with more than one variable?

Yes. As shown in the circle equation example, you can complete the square for each variable independently. This extends to ellipses, hyperbolas, and higher-dimensional quadratic forms. In multivariable calculus, completing the square for quadratic forms in two or more variables is a standard technique for classifying critical points.

What is the connection between completing the square and graphing parabolas?

The vertex form directly tells you the vertex and the direction the parabola opens (up if , down if ). The value of controls the width: larger values make the parabola narrower, smaller values make it wider. This makes completing the square the preferred method for graphing quadratics by hand.