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# Solving a Quadratic Equation Using Factoring

(KS4, Year 10)

## The Lesson

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 |

**(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 |

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.# 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**. We have solved the quadratic equation:

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

^{2}**x**:

^{2}Read more about solving a quadratic equation using factoring when the leading coefficient is not 1

## Factoring, Factorising

To write a quadratic equation as a product of two brackets is called 'to factor' or 'to factorise' the quadratic equation, depending on the country. The method is refered to as 'factoring' or 'factorising'.## Why the Method Works

Factoring is the opposite of expanding two brackets using the FOIL method. Let's go backwards. Start with the quadratic equation, whose roots are**x**and

_{1}**x**, factored into two brackets:

_{2}
(x + x

Expand the brackets using the FOIL method:
_{1})(x + x_{2})
x

We can see that the constant term is ^{2}+ (x_{1}+ x_{2})x + x_{1}x_{2}**x**and that the coefficient of the

_{1}x_{2}**x**term is

**x**.

_{1}+ x_{2}## Beware

## Be Careful with Signs 1

When a quadratic equation has been factored, the roots of the equation can be read off. But remember, you need to flip the sign. For instance, if the factored equation is...
(x + 2)(x + 3) = 0

...then the roots are:
x = −2 (−ve), x = −3 (−ve)

If the factored equation is...
(x + 2)(x − 3) = 0

...then the roots are:
x = −2 (−ve), x = 3 (+ve)

## Be Careful with Signs 2

Consider the quadratic equation shown below:
x

^{2}+ bx + c-
**b**is the coefficient of the**x**term. -
**c**is the constant term.

**b**and

**c**terms, the numbers you write in the brackets can be positive or negative. The image below defines what is meant by a positive or negative

**b**,

**c**and number in a bracket:

The following table gives a quick summary of what the signs must be:

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