Year 10: Solve the quadratic equation by factorisation

This cheat sheet will help you solve quadratic equations by factorisation. A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

What is Factorisation?

Factorisation involves rewriting the quadratic expression into the form (x + p)(x + q) = 0, where p and q are constants.

Steps to Solve by Factorisation

1. Rearrange the equation: Make sure the equation is in the form ax² + bx + c = 0.
2. Factorise the quadratic expression: Find two numbers that multiply to 'ac' (a multiplied by c) and add up to 'b'.
3. Rewrite the equation: Replace the 'bx' term with the sum of those two numbers.
4. Factorise the new expression: You should now be able to factorise the entire expression into two binomial factors.
5. Set each factor equal to zero: (x + p)(x + q) = 0 implies x + p = 0 and x + q = 0.
6. Solve for x: Solve each of the two linear equations to find the values of x.

Example

Solve: x² + 5x + 6 = 0

1. Find two numbers that multiply to 6 and add to 5. These are 2 and 3.
2. Rewrite the equation: x² + 2x + 3x + 6 = 0
3. Factorise: x(x + 2) + 3(x + 2) = 0
4. Rewrite: (x + 2)(x + 3) = 0
5. Solve: x + 2 = 0 => x = -2 and x + 3 = 0 => x = -3

Your solutions are x = -2 and x = -3.