What you'll learn
Solving quadratic equations by factorising is one of the main methods for finding the values of x that satisfy a quadratic. In this guide you will learn how to rearrange a quadratic equation to equal zero, how to factorise it, how to use the fact that a product equal to zero means one factor is zero, and how to interpret and check the solutions. This builds directly on factorising quadratic expressions.
Key terms and definitions
Quadratic equation — an equation of the form ax² + bx + c = 0.
Root / solution — a value of x that makes the equation true.
Factorise — write the quadratic as a product of two brackets.
Zero product property — if two things multiply to zero, at least one of them must be zero.
Rearrange — move all terms to one side so the equation equals zero.
Core concepts
Set the equation to zero
Before factorising, rearrange the equation so one side equals zero. For example, x² + 5x = −6 becomes x² + 5x + 6 = 0. All terms must be on one side, in the standard order ax² + bx + c = 0, before you factorise.
Factorise the quadratic
Factorise the quadratic expression into two brackets using the usual method (two numbers multiplying to c and adding to b, or a × c grouping for harder cases). For example, x² + 5x + 6 = (x + 2)(x + 3).
Use the zero product property
If a product equals zero, one of the factors must be zero. So if (x + 2)(x + 3) = 0, then either x + 2 = 0 or x + 3 = 0. Solve each: x = −2 or x = −3. A quadratic usually has two solutions, but sometimes a repeated root gives just one.
Solutions with a coefficient of x²
When the bracket contains a coefficient, set each factor to zero and solve carefully. For (2x − 1)(x + 4) = 0: 2x − 1 = 0 gives x = ½; x + 4 = 0 gives x = −4. So x = ½ or x = −4.
Checking solutions
Substitute each solution back into the original equation to check it gives zero (or both sides equal). This catches sign and arithmetic errors. Always present both solutions clearly.
Worked examples
Example 1: Basic solving
Solve x² + 7x + 10 = 0.
Factorise: (x + 2)(x + 5) = 0. So x + 2 = 0 or x + 5 = 0, giving x = −2 or x = −5.
Example 2: Rearranging first
Solve x² = 3x + 4.
Rearrange: x² − 3x − 4 = 0. Factorise: (x − 4)(x + 1) = 0. So x = 4 or x = −1.
Example 3: Coefficient of x²
Solve 2x² + 5x − 3 = 0.
Factorise: (2x − 1)(x + 3) = 0. So 2x − 1 = 0 (x = ½) or x + 3 = 0 (x = −3). x = ½ or x = −3.
Common mistakes and how to avoid them
Not setting the equation to zero first. Factorising only solves the equation when it equals zero.
Giving only one solution. A quadratic usually has two; state both.
Sign errors when solving each factor. x + 2 = 0 gives x = −2, not +2.
Mishandling coefficients. For 2x − 1 = 0, divide by 2 to get x = ½.
Skipping the check. Substitute back to confirm each solution.
Exam technique for Solving Quadratics by Factorising
Rearrange to equal zero in the form ax² + bx + c = 0.
Factorise accurately, taking out common factors first if possible.
Set each bracket to zero and solve separately.
Give both solutions clearly, including fractions where needed.
Check by substitution to catch errors.
Quick revision summary
To solve a quadratic equation by factorising, first rearrange it to equal zero (ax² + bx + c = 0), moving all terms to one side. Then factorise the quadratic into two brackets using the standard methods. Apply the zero product property: if two factors multiply to zero, at least one must be zero, so set each bracket equal to zero and solve. For example, x² + 7x + 10 = 0 → (x + 2)(x + 5) = 0 → x = −2 or x = −5. When a bracket has a coefficient, solve carefully (2x − 1 = 0 gives x = ½). A quadratic usually has two solutions, so always give both, and substitute them back to check. The common pitfalls are forgetting to set the equation to zero, giving only one root, sign slips when solving each factor, and not checking. Rearrange to zero, factorise, split into two equations, solve each, and verify — a reliable routine for every factorisable quadratic.