Completing the Square Calculator

Enter a quadratic expression with one or two variables (x, y) to complete the square and see the step-by-step procedure.

Quick Examples

Rate this tool

Instructions for Use

This online completing the square calculator is an algebraic tool designed to transform any quadratic expression, whether it has one or two variables (x and y). In addition to providing the exact final result, the solver generates the step-by-step analytical process to help you understand the underlying mathematical procedure.

Using the tool is simple through a single input field:

  1. Entering the expression: Type your polynomial expression (for example, 2x2 + 3x + 5) into the text box. The expression does not need to be arranged or simplified beforehand; the algebraic program will automatically group like terms. The system is flexible and accepts all types of coefficients: integers, decimals, fractions (rational numbers), and irrational values (like roots).
  2. Main result: When processing your input, the tool will display a highlighted box with the equivalent expression after completing the square. The solver works with exact precision; this means it will respect and preserve fractions and roots in their original symbolic form, avoiding any loss of information due to decimal rounding.
  3. Step-by-step solution: Just below the direct answer, a detailed explanation of the exercise will be displayed. You will be able to clearly see how the terms are grouped, and how the specific value needed to build a perfect square trinomial is calculated and introduced (by adding and subtracting it). Finally, the process will show you how that trinomial is factored to rewrite it as a binomial squared and how the remaining constant terms are simplified to arrive at the final expression.

Solved Exercises

The following are examples of problems solved by the calculator.

Complete the square for the expression x2 + 6x + 5.

Result

$$ \left(x + 3\right)^2 - 4 $$

Step-by-step solution

1. Identify the expression.

The expression we will work with is:

$$ x^2 + 6x + 5 $$

2. Prepare to complete the square.

We want to rewrite the expression to form a perfect square trinomial. To do this, we take half of the linear term's coefficient, square it, and then add and subtract that value so as not to change the original expression:

$$ x^2 + 6x + \left(\dfrac{6}{2}\right)^2 - \left(\dfrac{6}{2}\right)^2 + 5 \\[1.5em] x^2 + 6x + \left(3\right)^2 - \left(3\right)^2 + 5 \\[1.5em] x^2 + 6x + 9 - 9 + 5 $$

3. Factor and simplify.

We group the first three terms, which now form the perfect square trinomial \( x^2 \pm 2ax + a^2 \), and rewrite them as a binomial squared \( (x \pm a)^2 \).

$$ \left(x^2 + 6x + 9\right) - 9 + 5 \\[1.5em] \left(x^2 + 2(3)x + \left(3\right)^2\right) - 9 + 5 \\[1.5em] \left(x + 3\right)^2 - 9 + 5 $$

We simplify the constant terms:

$$ \left(x + 3\right)^2 - 4 $$
Complete the square for 2x2 + 3x + 6.

Result

$$ 2\left(x + \dfrac{3}{4}\right)^2 + \dfrac{39}{8} $$

Step-by-step solution

1. Identify the expression.

The expression we will work with is:

$$ 2x^2 + 3x + 6 $$

2. Factor out the leading coefficient.

We factor out the leading coefficient, applying it only to the terms containing the variable. We keep the constant term outside the parentheses:

$$ 2\left( x^2 + \dfrac{3}{2}x \right) + 6 $$

3. Prepare to complete the square.

We want to rewrite the expression to form a perfect square trinomial. To do this, we take half of the linear term's coefficient, square it, and then add and subtract that value so as not to change the original expression:

$$ 2\left( x^2 + \dfrac{3}{2}x + \left(\dfrac{\frac{3}{2}}{2}\right)^2 - \left(\dfrac{\frac{3}{2}}{2}\right)^2 \right) + 6 \\[1.5em] 2\left( x^2 + \dfrac{3}{2}x + \left(\dfrac{3}{4}\right)^2 - \left(\dfrac{3}{4}\right)^2 \right) + 6 \\[1.5em] 2\left( x^2 + \dfrac{3}{2}x + \dfrac{9}{16} - \dfrac{9}{16} \right) + 6 $$

4. Factor and simplify.

We group the first three terms, which now form the perfect square trinomial \( x^2 \pm 2ax + a^2 \), and rewrite them as a binomial squared \( (x \pm a)^2 \).

$$ 2\left( \left(x^2 + \dfrac{3}{2}x + \dfrac{9}{16}\right) - \dfrac{9}{16} \right) + 6 \\[1.5em] 2\left( \left(x^2 + 2\left(\dfrac{3}{4}\right)x + \left(\dfrac{3}{4}\right)^2\right) - \dfrac{9}{16} \right) + 6 \\[1.5em] 2\left( \left(x + \dfrac{3}{4}\right)^2 - \dfrac{9}{16} \right) + 6 $$

5. Distribute and simplify.

We distribute the factor outside the parentheses and simplify:

$$ 2\left(x + \dfrac{3}{4}\right)^2 - 2\left(\dfrac{9}{16}\right) + 6 \\[1.5em] 2\left(x + \dfrac{3}{4}\right)^2 - \dfrac{9}{8} + 6 \\[1.5em] 2\left(x + \dfrac{3}{4}\right)^2 + \dfrac{39}{8} $$
Complete the perfect square trinomial in the expression -5x2 + 4x - 8.

Result

$$ -5\left(x - \dfrac{2}{5}\right)^2 - \dfrac{36}{5} $$

Step-by-step solution

1. Identify the expression.

The expression we will work with is:

$$ -5x^2 + 4x - 8 $$

2. Factor out the leading coefficient.

We factor out the leading coefficient, applying it only to the terms containing the variable. We keep the constant term outside the parentheses:

$$ -5\left( x^2 - \dfrac{4}{5}x \right) - 8 $$

3. Prepare to complete the square.

We want to rewrite the expression to form a perfect square trinomial. To do this, we take half of the linear term's coefficient, square it, and then add and subtract that value so as not to change the original expression:

$$ -5\left( x^2 - \dfrac{4}{5}x + \left(\dfrac{\frac{4}{5}}{2}\right)^2 - \left(\dfrac{\frac{4}{5}}{2}\right)^2 \right) - 8 \\[1.5em] -5\left( x^2 - \dfrac{4}{5}x + \left(\dfrac{2}{5}\right)^2 - \left(\dfrac{2}{5}\right)^2 \right) - 8 \\[1.5em] -5\left( x^2 - \dfrac{4}{5}x + \dfrac{4}{25} - \dfrac{4}{25} \right) - 8 $$

4. Factor and simplify.

We group the first three terms, which now form the perfect square trinomial \( x^2 \pm 2ax + a^2 \), and rewrite them as a binomial squared \( (x \pm a)^2 \).

$$ -5\left( \left(x^2 - \dfrac{4}{5}x + \dfrac{4}{25}\right) - \dfrac{4}{25} \right) - 8 \\[1.5em] -5\left( \left(x^2 - 2\left(\dfrac{2}{5}\right)x + \left(\dfrac{2}{5}\right)^2\right) - \dfrac{4}{25} \right) - 8 \\[1.5em] -5\left( \left(x - \dfrac{2}{5}\right)^2 - \dfrac{4}{25} \right) - 8 $$

5. Distribute and simplify.

We distribute the factor outside the parentheses and simplify:

$$ -5\left(x - \dfrac{2}{5}\right)^2 + 5\left(\dfrac{4}{25}\right) - 8 \\[1.5em] -5\left(x - \dfrac{2}{5}\right)^2 + \dfrac{4}{5} - 8 \\[1.5em] -5\left(x - \dfrac{2}{5}\right)^2 - \dfrac{36}{5} $$
Rewrite by completing the squares in the two-variable formula x2 + y2 - 4x + 6y - 12.

Result

$$ \left(x - 2\right)^2 + \left(y + 3\right)^2 - 25 $$

Step-by-step solution

1. Identify the expression.

The expression we will work with is:

$$ x^2 - 4x + y^2 + 6y - 12 $$

2. Identify and group by variables.

The expression contains multiple variables. We group the terms corresponding to each variable in parentheses to work with them independently:

$$ \left( x^2 - 4x \right) + \left( y^2 + 6y \right) - 12 $$

3. Prepare to complete the squares.

We want to rewrite each group to form perfect square trinomials. To do this, we take half of the linear term's coefficient, square it, and then add and subtract that value within its respective group so as not to change the original expression:

$$ \left( x^2 - 4x + \left(\dfrac{4}{2}\right)^2 - \left(\dfrac{4}{2}\right)^2 \right) + \left( y^2 + 6y + \left(\dfrac{6}{2}\right)^2 - \left(\dfrac{6}{2}\right)^2 \right) - 12 \\[1.5em] \left( x^2 - 4x + \left(2\right)^2 - \left(2\right)^2 \right) + \left( y^2 + 6y + \left(3\right)^2 - \left(3\right)^2 \right) - 12 \\[1.5em] \left( x^2 - 4x + 4 - 4 \right) + \left( y^2 + 6y + 9 - 9 \right) - 12 $$

4. Factor the groups.

We group the first three terms for each variable, which now form perfect square trinomials \( v^2 \pm 2av + a^2 \), and rewrite them as binomials squared \( (v \pm a)^2 \).

$$ \left( \left(x^2 - 4x + 4\right) - 4 \right) + \left( \left(y^2 + 6y + 9\right) - 9 \right) - 12 \\[1.5em] \left( \left(x^2 - 2(2)x + \left(2\right)^2\right) - 4 \right) + \left( \left(y^2 + 2(3)y + \left(3\right)^2\right) - 9 \right) - 12 \\[1.5em] \left( \left(x - 2\right)^2 - 4 \right) + \left( \left(y + 3\right)^2 - 9 \right) - 12 $$

5. Simplify.

We remove the outer grouping parentheses and add the remaining constant terms:

$$ \left(x - 2\right)^2 - 4 + \left(y + 3\right)^2 - 9 - 12 \\[1.5em] \left(x - 2\right)^2 + \left(y + 3\right)^2 - 25 $$

Related Tools

Daniel Machado

Professor of Mathematics, graduated from the Faculty of Exact, Chemical and Natural Sciences of the National University of Misiones (UNAM). Developer and creator of RigelUp, dedicated to building tools for mathematical learning.

Last updated: (1 week ago)