Complete the Square Calculator

Turn a quadratic into a perfect square and use it to find vertex form or roots.

Drag & drop or upload an image or PDF

Enter a quadratic, such as x^2 + 6x - 7 = 0

Try an example

Build a perfect square trinomial

Completing the square rewrites a quadratic so part of it becomes (x + p)². That form makes the vertex visible and gives a direct route to solving by square roots.

When the leading coefficient is not 1, factor it from the x² and x terms first. For equations, add the same value to both sides to keep the balance.

How to use the complete the square calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

The completing term

Take half the x coefficient and square it to create a perfect square trinomial.

x^{2} + b x + (\frac{b}{2})^{2} = (x + \frac{b}{2})^{2}

A reliable way to work through it

Normalize the leading coefficient

Factor or divide so the coefficient of x² inside the working group is 1.

Add the completing term

Take half the x coefficient, square it, and keep an equation balanced.

Write the square

Replace the trinomial with its binomial square, then solve or read the vertex.

Worked example

Solve x² + 6x - 7 = 0 by completing the square

x^{2} + 6 x = 7

Move the constant to the other side.

x^{2} + 6 x + 9 = 16

Add (6/2)² = 9 to both sides.

(x + 3)^{2} = 16

Rewrite the perfect square trinomial.

x + 3 = \pm 4

Take both square roots.

x = 1 or x = - 7

Isolate x in both cases.

The roots are x = 1 and x = -7.

Common mistakes to check

Using b² instead of (b/2)²

The completing term is the square of half the x coefficient.

Ignoring the leading coefficient

The shortcut assumes the coefficient of x² is 1 inside the group.

Taking only the positive square root

Solving a squared equation requires both positive and negative roots.

Quadratic Equation CalculatorFind real or complex roots and see which quadratic method fits the equation.Factoring CalculatorBreak a polynomial into factors using the pattern that fits, then verify by expanding.Polynomial CalculatorWork with polynomial operations, factors, roots, and remainders in one place.Solve for X CalculatorEnter an equation and see how to isolate x, simplify both sides, and check the result.

Questions students ask

Can the calculator convert a quadratic to vertex form?

Yes. Completing the square rewrites y = ax² + bx + c as y = a(x - h)² + k.

Does it work when the leading coefficient is not 1?

Yes. The coefficient is factored from the variable terms before the completing term is added.

Why complete the square instead of factoring?

It works when integer factoring is difficult and also reveals the vertex of the parabola.

How is completing the square related to the quadratic formula?

The quadratic formula can be derived by completing the square on the general equation ax² + bx + c = 0.