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
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

Find the pairs of numbers that multiply to make 4.

4 is the constant term in the quadratic equation.
4 = 1 × 4 4 = 2 × 2
Don't forget: We have found the factors of 4.

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: x2 + 5x + 4 = (x + 1)(x + 4) = 0. We have solved the quadratic equation: x = −1, x = −4.

Lesson Slides

The slider below shows another real example of how to solve a quadratic equation using factoring. In this example, there are minus signs (−) in the quadratic equation. Open the slider in a new tab

Solving a Quadratic Equation Using Factoring When There Is a Number in Front of the x2

In all the examples in this lesson, there has been no number in front of the x2. (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 x2:

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 x1 and x2, factored into two brackets:
(x + x1)(x + x2)
Expand the brackets using the FOIL method:
x2 + (x1 + x2)x + x1x2
We can see that the constant term is x1x2 and that the coefficient of the x term is x1 + x2.

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:
x2 + bx + c
  • b is the coefficient of the x term.
  • c is the constant term.
Depending on the sign of the 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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See Also

What is the FOIL method? What is a factor? What is a constant? What is a coefficient?