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:
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 | ⇒ 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: x2 + 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 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:
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.

