The LessonFactoring (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 x2 term) is not 1. 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 to Solve Quadratic Equations Using Factoring (When the Leading Coefficient is Not 1)Solving a quadratic equation using factoring is easy.
QuestionSolve the quadratic equation shown below using factoring.
Multiply the coefficent of the x2 term (2) with the constant (6).
2 × 6 = 12
Find the pairs of numbers that multiply to make the answer (12).Don't forget: We have found the factors of 12.
12 = 1 × 12
12 = 2 × 6
12 = 3 × 4
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.3 and 4 add up to make 7.
1 + 12 = 13 ✖
2 + 6 = 8 ✖
3 + 4 = 7 ✔
Rewrite 7x as a sum of two x-terms, using the pairs of factors found in Step 3.
Group the terms for factoring.
Factor each bracket by taking the greatest common factor out.
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.
Equate the first bracket to 0 and solve to find x.
x + 2 = 0 ⇒ x = −2
Equate the second bracket to 0 and solve to find x.
2x + 3 = 0 ⇒ x = −3⁄2
Answer:We have factored the quadratic equation: 2x2 + 7x + 6 = (x + 2)(2x + 3) = 0. We have solved the quadratic equation: x = −2, x = −3⁄2.
Lesson SlidesThe slider below shows another real example of how to solve 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 country. The method is referred to as 'factoring' or 'factorising'.
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)(2x + 3) = 0...then the roots are:
x = −2 (−ve), x = −3⁄2 (−ve)If the factored equation is...
(x + 2)(2x − 3) = 0...then the roots are:
x = −2 (−ve), x = 3⁄2 (+ve)
Be Careful with Signs 2Consider the quadratic equation shown below:
ax2 + bx + c
- b is the coefficient of the x term.
- c is the constant term.