Free maths tool · Level: Ages 14 to 16
Factoring quadratics
Slide the two factors and watch the rectangle. Its four pieces together are exactly the expanded form, and that is where the rule comes from.
Factoring means rewriting a sum as a product. For x² + bx + c you look for two numbers that add to b and multiply to c, and then x² + bx + c is (x + p)(x + q). For x² + 5x + 6 those are 2 and 3, because 2 + 3 = 5 and 2 · 3 = 6, so you write (x + 2)(x + 3).
Why sum and product works
The rectangle is (x + p) wide and (x + q) tall. Its area can be written two ways: length times width, or the sum of the four pieces it is cut into. Those two have to agree, and that agreement is the factorisation.
Look at the two long pieces. Both are x long, and together they are p + q wide, so together they come to (p + q)·x. The corner is p by q, so p · q. That is where the rule comes from: the middle number is the sum, the last one is the product.
Take out a common factor first
Before reaching for sum and product, always check whether every term shares a factor. In 2x² + 10x + 12 there is a 2 throughout, so take it outside first: 2(x² + 5x + 6). Only then factor what is left inside the brackets.
Skipping that step is the most common mistake in this topic. You still get a factorisation, but not the complete one, and in a test that costs marks.
When sum and product will not work
Sometimes no pair of whole numbers has both the right sum and the right product. The expression then does not factor over the integers, and you move on to the discriminant: it gives the roots, and with those the factors are a(x - x₁)(x - x₂).
If there is a minus, at least one of the two numbers is negative. For x² - 5x + 6 take 2 and 3 with both signs negative, since -2 + -3 = -5 and -2 · -3 = 6. For x² + x - 6 the signs differ: 3 and -2.
Is factoring still guesswork?
A tutor practises the method on your own exercises, until you spot the numbers straight away.
Frequently asked questions
- What does factoring mean?
- Rewriting a sum as a product. Instead of x² + 5x + 6 you write (x + 2)(x + 3). That is useful because a product is zero as soon as one of its factors is, which hands you the solutions.
- What is the method for factoring?
- Take out a common factor first if there is one. Then check whether it is a special product such as a² - b². Failing that, look for two numbers with sum b and product c. If that fails too, use the discriminant.
- How do you factor when there is a minus sign?
- The product tells you whether the signs match. If c is positive both numbers share a sign, and b says which. If c is negative the signs differ, and the larger number takes the sign of b.
- What if the coefficient of x² is not 1?
- Take it outside if you can. If you cannot, as in 2x² + 7x + 3, use the discriminant to find the roots and write a(x - x₁)(x - x₂).
- Why is there a minus in the factor for a positive root?
- Because (x - 3) is zero when x is 3. The factor and the root always carry opposite signs: (x + 2) belongs to the root -2. That is the sign slip made most often.