How do you solve a quadratic equation by factorising?

The rule everything rests on

If a product of two numbers equals zero, at least one of them is zero. There is no other way to get zero from multiplication, and this is unique to zero — a product equal to six tells you nothing comparable.

So the form to aim for is a product on one side and zero on the other. From there each factor is set to zero separately, and each of the resulting equations is linear.

Why everything moves to one side

Without zero on the other side the rule simply does not apply. If a product equals six, each factor could be almost anything, and nothing follows about either of them.

That is why the first step is always the same: collect everything on one side, arrange into standard form, and only then factorise. Skipping it is the single most common error in this topic.

How many solutions to expect

Usually two, one from each factor. If the two factors are identical — that is, the expression is a perfect square — there is exactly one solution, called a double root.

An equation with no constant term factorises by taking out x, and then one of the solutions is zero. Zero is a solution like any other, and dividing through by x erases it — a mistake worth knowing about in advance.

Worked examples

1. Solve x squared plus 5x plus 6 equals zero

   1. The right side is already zero
   2. Factorise: two numbers with product 6 and sum 5, namely 2 and 3
   3. This gives (x + 2)(x + 3) equals zero
   4. Set each factor to zero separately

   Answer: x equals minus 2, or x equals minus 3

2. Solve x squared equals 4x

   1. Move everything to one side: x squared minus 4x equals zero
   2. Take the common factor: x times (x − 4) equals zero
   3. Set each factor to zero

   Answer: x equals 0 or x equals 4. Dividing by x would have lost the zero

3. Solve x squared minus 6x plus 9 equals zero

   1. Recognise a perfect square: this is (x − 3) squared
   2. A factor multiplied by itself equals zero
   3. So x minus 3 equals zero

   Answer: x equals 3, a single, double root

Common mistakes

Factorising without moving everything to one side

The zero product rule applies only when the other side is zero. Factorising against some other number and setting each factor equal to it gives answers that are not solutions of the equation at all.

Dividing both sides by x

If x might be zero, dividing erases a solution. Take x out as a factor instead of dividing by it, and both solutions survive.

Flipping the sign in the solution

If x plus two equals zero, then x is minus two. The bracket shows a plus and the solution is a minus, and that swap happens fast when people skip writing the small equation down.

More in Quadratic Equations

• How do you expand brackets and what are the special products?
• How do you factorise an algebraic expression?
• What is the quadratic formula and how do you use it?
• What is the discriminant and how many solutions does an equation have?
• How do you find the vertex and the intercepts of a parabola?