Year 10: Solve the quadratic equation by factorisation

This cheat sheet will guide you through solving quadratic equations by factorisation. A quadratic equation has the general form: $a{x}^2+bx+c=0$, where a, b, and c are constants and a ≠ 0.

What is factorisation?

Factorisation involves expressing the quadratic expression as a product of two linear expressions (factors).

Steps to solve by factorisation

1. Rearrange the Equation: Make sure the equation is in the standard form: $a{x}^2+bx+c=0$
2. Factorise: Find two numbers that multiply to 'ac' (a times c) and add up to 'b'. Use these numbers to split the middle term ('bx') into two terms.
3. Factor out the Common Factor: Factor out the common factor from the two new terms.
4. Set each factor to zero: Set each of the linear factors equal to zero.
5. Solve for x: Solve each of the resulting linear equations for 'x'. These are your solutions to the quadratic equation.

Example

Solve: $x^2+5x+6=0$

1. We have the equation: $x^2+5x+6=0$

2. Find two numbers that multiply to 6 and add to 5. These are 2 and 3.

3. Rewrite the middle term: $x^2+2x+3x+6=0$

4. Factor out common factors: $x(x+2)+3(x+2)=0$ (Incorrect - needs adjustment)

4. Correct Factorisation: $(x+2)(x+3)=0$

5. Set each factor to zero: $x+2=0$ or $x+3=0$

6. Solve: $x=-2$ or $x=-3$

These are the solutions to the equation.