# Solve by Completing the Square Calculator + Online Solver With Free Steps

The **Solve by completing the Square Calculator** is used to solve a quadratic equation by using the complete square method. It takes a **quadratic equation** as input and outputs the solutions for the quadratic equation using the completing square method.

A quadratic polynomial is a **second-degree** polynomial. The quadratic equation can be written in the form given below:

**$p x^2$ + q x + r = 0 **

Where p, q and r are the coefficients of $x^2$, x and $x^0$, respectively. If $p$ is equal to zero, the equation becomes linear.

The completing square method is one of the methods to solve the quadratic equation. The other methods include **factorization** and using the **quadratic formula**.

The completing square method uses the two **formulas** to form a complete square of the quadratic equation. The two formulas are given below:

\[ {(a + b)}^2 = a^2 + 2ab + b^2 \]

\[ {(a \ – \ b)}^2 = a^2 \ – \ 2ab + b^2 \]

The calculator adds or subtracts numerical values to form the complete squares of the quadratic equation.

## What Is a Solve by Completing the Square Calculator?

**The Solve by Completing the Square Calculator is an online tool that solves the quadratic equation by using the square completion method.**

It changes the quadratic equation into a complete square form and provides the solutions for the unknown variable.

The **input equation** should be of the form $p x^2$ + q x + r = 0 where p should not be equal to zero for the equation to be quadratic.

## How To Use the Solve by Completing the Square Calculator

The user can follow the steps given below to solve a quadratic equation by using the Solve by Completing the Square Calculator

### Step 1

The user must first enter the quadratic equation in the input tab of the calculator. It should be entered in the block, “**Quadratic Equation**”. The quadratic equation is an equation with degree two.

For the** default** example, the calculator inputs the quadratic equation given below:

**$x^{2}$ – x – 3 = 0 **

If an equation with a **degree** **greater **than** two** is entered in the calculator’s input window, the calculator prompts “Not a valid input; please try again”.

### Step 2

The user must press the button labeled, “**Solve by Completing the Square**” for the calculator to process the input quadratic equation.

### Output

The calculator solves the quadratic equation by completing the square method and displays the output in the **three windows** given below:

#### Input Interpretation

The calculator interprets the input and displays “**complete the square**” along with the input equation in this window. For the **default** example, the calculator shows the input interpretation as follows:

**complete the square = $x^{2}$ – x – 3 = 0 **

#### Results

The calculator solves the quadratic equation by using the completing square method and displays the **equation** in this window.

The calculator also provides all the **mathematical steps** by clicking on “Need a step-by-step solution for this problem?”.

It processes the input equation to check if the left-hand side of the equation forms a complete square.

Adding and subtracting $ { \left( \frac{1}{2} \right) }^{2}$ in the left-hand side of the equation to form a complete square.

\[ \Big\{ (x)^2 \ – \ 2(x) \left( \frac{1}{2} \right) + { \left( \frac{1}{2} \right) }^{2} \Big\} \ – \ { \left( \frac{1}{2} \right) }^{2} \ – \ 3 = 0 \]

\[ { \left( x \ – \ \frac{1}{2} \right) }^{2} \ – \ \frac{1}{4} \ – \ 3 = 0 \]

\[ { \left( x \ – \ \frac{1}{2} \right) }^{2} \ – \ \frac{1-12}{4} = 0 \]

The Result window shows the equation given below:

\[ { \left( x \ – \ \frac{1}{2} \right) }^{2} \ – \ \frac{13}{4} = 0 \]

#### Solutions

After using the completing square method, the calculator **solves** the quadratic equation for the value of $x$. The calculator displays the solution by solving the equation given below:

\[ { \left( x \ – \ \frac{1}{2} \right) }^{2} \ – \ \frac{13}{4} = 0 \]

Adding $ \frac{13}{4}$ on both sides of the equation gives:

\[ { \left( x \ – \ \frac{1}{2} \right) }^{2} \ – \ \frac{13}{4} + \frac{13}{4} = \frac{13}{4} \]

\[ { \left( x \ – \ \frac{1}{2} \right) }^{2} = \frac{13}{4} \]

Taking square root on both sides of the equation gives:

\[ x \ – \ \frac{1}{2} = \pm \frac{ \sqrt{13} }{2} \]

The Solutions window shows the solution for $x$ for the default example as follows:

\[ x = \frac{1}{2} \ – \ \frac{ \sqrt{13} }{2} \]

## Solved Examples

The following examples are solved through the Solve by Completing the Square Calculator

### Example 1

Find the roots of the quadratic equation:

**$x^{2}$ + 6x + 7 = 0 **

By using the **completing square method**.

### Solution

The user must first enter the **quadratic equation** $x^{2}$ + 6x + 7 = 0 in the input tab of the calculator.

After pressing the “Solve by Completing the Square” button, the calculator shows the **input interpretation** as follows:

**Complete the square = $x^{2}$ + 6x + 7 = 0 **

The calculator uses the complete square method and re-writes the equation in the form of the complete square. The **Result** window shows the following equation:

**${( x + 3 )}^2$ – 2 = 0 **

The **Solutions** window shows the value of $x$ which is given below:

**x = – 3 – $\sqrt{2}$**

### Example 2

By using the **completing square method**, find the roots of the equation given as:

**$x^2$ + 8x + 2 = 0 **

### Solution

The **quadratic equation** $x^2$ + 8x + 2 = 0 must be entered in the calculator’s input window. After submitting the input equation, the calculator shows the **input interpretation** as follows:

**Complete the square = $x^{2}$ + 8x + 2 = 0 **

The **Results** window shows the above equation after performing the completing square method. The equation becomes:

**${( x + 4 )}^2$ – 14 = 0 **

The calculator displays the **solution** for the above quadratic equation as follows:

**x = – 4 – $\sqrt{14}$**