Solving a Quadratic Equation Using Factoring
(KS4, Year 10)
The LessonFactoring (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 EquationsWhen 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 FactoringSolving a quadratic equation using factoring is easy.
QuestionSolve the quadratic equation shown below using factoring.
Write each of these numbers (1 and 4) being added to x in a bracket, all equal to 0.
Equate the first bracket to 0 and solve to find x.
x + 1 = 0 ⇒ x = −1
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 SlidesThe 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 x2In 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, FactorisingTo 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 WorksFactoring 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 + x1x2We can see that the constant term is x1x2 and that the coefficient of the x term is x1 + x2.
Be Careful with Signs 1When 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 2Consider the quadratic equation shown below:
x2 + bx + c
- b is the coefficient of the x term.
- c is the constant term.
The following table gives a quick summary of what the signs must be:
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