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# Solving a Quadratic Equation Using Factoring

(When the Leading Coefficient is Not 1)

(KS4, Year 10)

## The Lesson

Factoring (or factorising) is a way of simplifying a quadratic equation. In this lesson, we will look at quadratic equations where the leading coefficient (the number in front of the**x**term) is not 1. Factoring a quadratic equation writes it as two brackets multiplying each other.

^{2}Factoring is the opposite of expanding the two brackets out using the FOIL method.

## How to Solve Quadratic Equations Using Factoring (When the Leading Coefficient is Not 1)

Solving a quadratic equation using factoring is easy.## Question

Solve the quadratic equation shown below using factoring.## Step-by-Step:

# 1

# 2

Find the pairs of numbers that multiply to make the answer (

**12**).
12 = 1 × 12
12 = 2 × 6
12 = 3 × 4

**Don't forget:**We have found the factors of**12**.# 3

Look at the pairs of factors found in

**Step 2**. Do any of them add up to**7**?**7**is the coefficient of the**x**term in the quadratic equation.
1 + 12 = 13 ✖
2 + 6 = 8 ✖
3 + 4 = 7 ✔

**3**and**4**add up to make 7.# 4

Rewrite

**7x**as a sum of two**x**-terms, using the pairs of factors found in**Step 3**.# 5

# 6

# 7

Each term in brackets should be the same.
If it is not, go back to

**Step 5**and regroup the terms. If the terms in the brackets are the same, we can group the terms outside the brackets into their own bracket.# 8

Equate the first bracket to 0 and solve to find

**x**.
x + 2 = 0 ⇒

**x = −2**# 9

Equate the second bracket to 0 and solve to find

**x**.
2x + 3 = 0 ⇒

**x = −**^{3}⁄_{2}## Answer:

We have factored the quadratic equation:**2x**. We have solved the quadratic equation:

^{2}+ 7x + 6 = (x + 2)(2x + 3) = 0**x = −2, x = −**.

^{3}⁄_{2}## 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 country. The method is referred to as 'factoring' or 'factorising'.## 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)(2x + 3) = 0

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

If the factored equation is...
^{3}⁄_{2}(−ve)
(x + 2)(2x − 3) = 0

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

^{3}⁄_{2}(+ve)## Be Careful with Signs 2

Consider the quadratic equation shown below:
ax

^{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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