A quadratic equation contains up to . There are many ways to solve quadratics. All quadratic equations can be written in the form where , and are numbers ( cannot be equal to 0, but and can be).

Here are some examples of quadratic equations in this form:

• . , and • . , and (in this example, the bracket can be expanded to )
• . , and (this will expand to )
• . , and • . , and (in this example , but has been rearranged to the other side of the equation) is an example of a quadratic equation that can be solved simply.

Divide both sides by 3

## Solving by factorising

If the of two numbers is zero then one or both of the numbers must also be equal to zero. Therefore if , then and/or .

If , then or , or both. Factorising quadratics will also be used to solve these equations.

Expanding the brackets gives , which simplifies to . is the reverse process of expanding brackets, so factorising gives .

### Example

Solve .

The product of and is 0, so or , or both.  or .

Question

Solve .

The product of and is 0, so one or both brackets must also be equal to 0.   or Question

Solve .

The quadratic will be in the form .

Find two numbers with a product of 12 and a sum of 7. , and , so and are equal to 3 and 4. This gives: The product of and is 0, so or , or both.   or 