# Completing The Square Formula

Completing the square formula is a technique or method to convert a quadratic polynomial or equation into a perfect square with some additional constant. A quadratic expression in variable x: ax^{2} + bx + c, where a, b and c are any real numbers but a ≠ 0, can be converted into a perfect square with some additional constant by using completing the square formula or technique.

**Note:** Completing the square formula is used to derive the quadratic formula.

## What Is Completing the Square Formula?

Completing the square formula is a technique or method to find the roots of the given quadratic equations. A quadratic equation in its standard form, in variable x, is ax^{2} + bx + c = 0, where a, b and c are any real numbers but a ≠ 0.

### Completing the Square Formula:

The formula for completing the square is :

\(ax^2 + bx + c \Rightarrow ( {x + m} )^2 + {\rm{ constant}}\)

where, m is any real number.

Let us understand the completing the square formula using solved examples.

## Examples Using Completing the Square Formula

**Example 1: Use completing the square method to solve: x ^{2} - 4x - 8 = 0.**

**Solution:**

Let’s transpose the constant term to the other side of the equation:

x^{2} – 4x = 8

Now, take half of the coefficient of the x-term, which is –4, including the sign, which gives –2. Square –2 to get +4, and add this squared value to both sides of the equation:

x^{2} – 4x + 4 = 8 + 4

x^{2} – 4x + 4 = 12

This process creates a quadratic expression that is a perfect square on the left-hand side of the equation. Simply we can replace the quadratic with the squared-binomial form:

(x – 2)^{2} = 12

Now, we've completed the expression to create a perfect-square binomial, let’s solve:

\(\begin{array}{l}

( {x - 2} )^2 = 12 \\

( {x - 2} ) = \pm \sqrt {12} \\

x - 2 = \pm 2\sqrt 3 \\

x = 2 \pm 2\sqrt 3 \\

\end{array}\)

**Answer:** **Using completing the square method, \(x = 2 \pm 2\sqrt 3 \).**

**Example 2: Use completing the square method to solve: x ^{2} + 4x - 5 = 0.**

**Solution:**

Let’s transpose the constant term to the other side of the equation:

x^{2} + 4x = 5

Now, take half of the coefficient of the x-term, which is –4, including the sign, which gives –2. Square –2 to get +4, and add this squared value to both sides of the equation:

x^{2} + 4x + 4 = 5 + 4

x^{2} – 4x + 4 = 9

This process creates a quadratic expression that is a perfect square on the left-hand side of the equation. Simply we can replace the quadratic with the squared-binomial form:

(x – 2)^{2} = 9

Now, we've completed the expression to create a perfect-square binomial, let’s solve:

\(\begin{array}{l}

( {x - 2} )^2 = 9 \\

( {x - 2} ) = \pm \sqrt {9} \\

x - 2 = \pm 3 \\

x = 2 \pm 3 \\

x = 5, - 1

\end{array}\)

**Answer:** **Using completing the square method, \(x = 5, - 1\).**

**Example 3: ****Which constant must be added and subtracted to solve the given quadratic equation by the method of completing the square: x ^{2 }+ 16x − 17 = 0?**

**Solution:**

Solution by completing the square for:

𝑥^{2 }+ 16𝑥 − 17 = 0

Keep 𝑥 terms on the left and move the constant to the right side by adding it on both sides

𝑥^{2 }+ 16𝑥 = 17

Take half of the 𝑥 term and square it

[16×(1/2)]^{2 }= 64

**Answer: We need to add and subtract 64 on both sides to solve the given equation using completing the square formula. **

## FAQs on Completing the Square Formula

### What Is Completing the Square Formula?

Completing the square is the formula required to convert a quadratic polynomial or equation into a perfect square with some additional constant. It is given as,

\(ax^2 + bx + c \Rightarrow ( {x + m} )^2 + {\rm{ constant}}\), where, m is any real number.

### How Do You Know When to Apply Complete the Square Formula?

Completing the square formula is used to represent a quadratic polynomial or equation into a perfect square with some additional constant. It is thus used when we have to factorize any quadratic polynomial. We apply converting the square formula to convert a quadratic expression of the form ax^{2} + bx + c to the vertex form a(x−h)^{2 }+ ka(x−h)^{2 }+ k, we complete the square.

### What Is the Use of Completing the Square Formula?

Completing the square formula is used when we want to represent a quadratic polynomial or equation into a perfect square with some additional constant and thus used to factorize a quadratic polynomial.