# How to Solve a Quadratic Equation Using Factoring

## Solving a Quadratic Equation Using Factoring

Factoring (or factorising) is a way of simplifying a quadratic equation.

Factoring a quadratic equation writes it as two brackets multiplying each other:

Factoring is the opposite of expanding the two brackets out using the FOIL method.

## How Factoring Solves Quadratic Equations

When two quantities are multiplied to make 0, this means that either or both of them are equal to 0.

A × B = 0 | ⇒ A = 0, B = 0 |

When a quadratic equation is factored, it consists of two brackets multiplying each other to equal 0.

For example, **(x + 1)(x + 4) = 0** is a factored quadratic equation.

Either or both of the brackets are equal to 0:

(x + 1) × (x + 4) = 0 | ⇒ x + 1 = 0, x + 4 = 0 |

We can find the roots of the quadratic equation:

x + 1 = 0 ⇒ x = −1

x + 4 = 0 ⇒ x = −4

## How to Solve Quadratic Equations Using Factoring

Solving a quadratic equation using factoring is easy.

### Question

Solve the quadratic equation shown below using factoring.

### Step-by-Step:

# 1

# 2

Look at the pairs of factors found in **Step 1**.

Do any of them add up to **5**?

**5** is the coefficient of the **x** term in the quadratic equation.

1 + 4 = 5 ✔

2 + 2 = 4 ✖

**1** and **4** add up to make 5.

# 3

Write each of these numbers (1 and 4) being added to **x** in a bracket, all equal to 0.

We have factored the quadratic equation.

Check you have factored the quadratic equation correctly by expanding the brackets using the FOIL method and seeing if you get back to the original equation.

Now lets use the factored quadratic equation to solve the quadratic equation.

# 4

Equate the first bracket to 0 and solve to find **x**.

x + 1 = 0 ⇒ **x = −1**

# 5

Equate the second bracket to 0 and solve to find **x**.

x + 4 = 0 ⇒ **x = −4**

### Answer:

We have factored the quadratic equation:

**x ^{2} + 5x + 4 = (x + 1)(x + 4) = 0**.

We have solved the quadratic equation:

**x = −1, x = −4**.

## Solving a Quadratic Equation Using Factoring When There Is a Number in Front of the x^{2}

In all the examples in this lesson, there has been no number in front of the **x ^{2}**. (In fact, there is a coefficient of 1, which does not need to be written).

You will need to learn how to factor a quadratic equation when there is a number that is not equal to 1 in front of **x ^{2}**:

Read more about solving a quadratic equation using factoring when the leading coefficient is not 1