Solving quadratics by factorising

If two things multiply to make 0, at least one of them must be 0. So once a quadratic equation is factorised, set each bracket equal to 0.

1. Make sure one side is 0.
2. Factorise the other side.
3. Set each bracket equal to 0 and solve.

A quadratic equation usually has two solutions (also called roots).

Example: solve x² + 2x − 15 = 0.

1. Factorise: (x + 5)(x − 3) = 0
2. x + 5 = 0, so x = −5
3. x − 3 = 0, so x = 3
4. x = −5 or x = 3

You can only use the "each bracket = 0" rule when one side is 0. Move everything to one side first.

Example: solve x² = 5x + 14.

1. Subtract 5x and 14 from both sides: x² − 5x − 14 = 0
2. Factorise: (x − 7)(x + 2) = 0
3. x = 7 or x = −2

Example: solve x² = 6x.

1. x² − 6x = 0
2. Take out the common factor x: x(x − 6) = 0
3. x = 0 or x = 6

Example: solve x² = 49.

1. Square root both sides, remembering the negative root.
2. x = 7 or x = −7

Factorise using the split-the-middle method, then set each bracket to 0. The solutions can be fractions.

Example: solve 2x² + 7x + 3 = 0.

1. Factorise: (2x + 1)(x + 3) = 0
2. 2x + 1 = 0, so 2x = −1 and x = −½
3. x + 3 = 0, so x = −3
4. x = −½ or x = −3

Harder example: solve 3x² − 10x = 8.

1. Rearrange: 3x² − 10x − 8 = 0
2. Factorise: (3x + 2)(x − 4) = 0
3. 3x + 2 = 0, so x = −2/3
4. x = −2/3 or x = 4

Problems about areas often lead to a quadratic equation.

1. Write the quantities in terms of x.
2. Form an equation and rearrange it to = 0.
3. Solve, then reject any solution that doesn't make sense, such as a negative length.

Example: a rectangle is x cm wide and (x + 5) cm long. Its area is 36 cm². Find its width.

1. Area = width × length: x(x + 5) = 36
2. Expand and rearrange: x² + 5x − 36 = 0
3. Factorise: (x + 9)(x − 4) = 0, so x = −9 or x = 4
4. A width can't be negative, so the width is 4 cm

Check: 4 × 9 = 36.