How to factor a quadratic equation (factorising quadratics)

To factor a quadratic equation like x² + 7x + 12, find two numbers that multiply to the constant term (12) and add to the middle coefficient (7). Here they are 3 and 4, so x² + 7x + 12 = (x + 3)(x + 4). When the x² term has a coefficient other than 1, as in 2x² + 7x + 6, use the ac method: multiply a and c, find the pair for that product, split the middle term and group. Always take out any common factor first, and check your answer by expanding the brackets. In the UK this is called factorising quadratics.

What factoring a quadratic means

A quadratic expression has the form ax² + bx + c. Factoring it means writing it as a product, usually of two brackets:

Factoring is the reverse of expanding (multiplying out). Expanding (x + 3)(x + 4) gives x² + 4x + 3x + 12 = x² + 7x + 12; factoring takes you back. The Common Core standards for US high school algebra put it this way: factor a quadratic expression to reveal the zeros of the function it defines. AQA's GCSE Maths specification includes factorising x² + bx + c, including the difference of two squares, and, in its Higher content, ax² + bx + c.

How to factor a quadratic: which method?

Method 1: take out a common factor

Look for a number or letter that divides every term, and take it outside a bracket. In 4x² + 12x, both terms divide by 4x:

Do this first, every time. It makes the numbers smaller and can turn a hard quadratic into an easy one: 3x² − 3x − 36 = 3(x² − x − 12) = 3(x − 4)(x + 3).

Method 2: factoring x² + bx + c

When the x² term has no number in front (a = 1), you need two numbers that:

• multiply to give c, the constant term, and
• add to give b, the coefficient of x.

Then x² + bx + c = (x + first number)(x + second number).

Example 1: both signs positive

Factor x² + 7x + 12. List the factor pairs of 12 and their sums:

3 and 4 add to 7, so:

Many teachers draw this search as a diamond, or X: the product goes on top, the sum at the bottom, and the two numbers you are looking for go on the sides.

Example 2: a negative constant

Factor x² + 2x − 15. The product is negative, so one number is positive and one is negative. The sum is +2, so the bigger one is positive. The pair is +5 and −3:

Example 3: a negative middle term

Factor x² − 9x + 20. The product is positive and the sum is negative, so both numbers are negative: −4 and −5.

The sign rules

Now try one. Factor the quadratic, then press Mark to check it in the course.

Method 3: the ac method, for ax² + bx + c

When a is not 1, the quick pair search no longer works on its own. The ac method, also called splitting the middle term or factoring by grouping, always does.

1. Multiply a by c.
2. Find two numbers that multiply to ac and add to b.
3. Use them to split the middle term into two terms.
4. Factor the first two terms and the last two terms separately.
5. Take out the bracket they share.

Example 4: 2x² + 7x + 6

Here a = 2 and c = 6, so ac = 12. The pair that multiplies to 12 and adds to 7 is 3 and 4. Split 7x into 3x + 4x:

Example 5: with negatives, 6x² − x − 12

ac = 6 × (−12) = −72. You need a pair that multiplies to −72 and adds to −1: that is 8 and −9.

If the two brackets after grouping do not match, check the signs: when you take a negative number out of the second pair, the signs inside the bracket flip.

The box method

Some people prefer a grid, called the box method or area model. Put the x² term and the constant in opposite corners, the two split terms in the other corners, then find what each row and column has in common. The grid for x² + 7x + 12 looks like this:

Method 4: difference of two squares

When a quadratic is one square minus another, with no x term, use:

So x² − 49 = (x + 7)(x − 7), and 9x² − 25 = (3x + 5)(3x − 5). It works because the middle terms cancel when you expand: +5·3x and −5·3x. A sum of two squares, such as x² + 49, does not factor with real numbers.

Method 5: perfect square trinomials

If the first and last terms are squares and the middle term is twice their product, the quadratic is a perfect square:

Check: 25 = 5², and the middle term is 2 × 5 × x = 10x. The same pattern with a minus sign gives x² − 10x + 25 = (x − 5)².

Method 6: when a quadratic will not factor

Many quadratics do not factor with whole numbers. To check before you waste time, work out the discriminant, b² − 4ac. If it is a perfect square (0, 1, 4, 9, 16 and so on), the quadratic factors over the integers. If not, it does not.

For x² + 3x + 5, b² − 4ac = 9 − 20 = −11, which is not a perfect square, so there is no factorization. Use the quadratic formula or completing the square instead. Our guide to quadratic equations covers all four ways to solve one, and the completing the square guide works through that method.

Solving a quadratic equation by factoring

To solve a quadratic equation, get 0 on one side, factor, then use the zero product property: if two things multiply to give 0, at least one of them is 0.

So x + 5 = 0 or x − 3 = 0, giving x = −5 or x = 3. These are the zeros of the function, where its graph crosses the x-axis. For 2x² + 7x + 6 = 0, the factors (x + 2)(2x + 3) give x = −2 or x = −3/2.

The Factoring Calculator factors any quadratic or polynomial you type in and shows which method it used, so you can check your working.

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Common mistakes

• Forgetting the common factor. 2x² + 14x + 24 should start as 2(x² + 7x + 12), then 2(x + 3)(x + 4).
• Getting the signs the wrong way round. (x − 5)(x + 3) expands to x² − 2x − 15, not x² + 2x − 15. Always expand to check.
• Using the pair search when a is not 1. For 2x² + 7x + 6, 3 and 4 do not go straight into the brackets; use the ac method.
• Solving without 0 on one side. (x + 1)(x − 2) = 4 does not mean x + 1 = 4. Expand, rearrange to = 0, then factor again.
• Factoring a sum of squares. x² + 9 has no real factors.

Practise factoring in a short timed test

Pre-Algebra & Algebra 1 and Geometry, Algebra 2 & SAT Math each give you a short timed practice test every day, with 8 questions worth 25 to 35 marks across the course, so factoring turns up regularly but not in every test.

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