# Year 9 Maths Chapter 8 - Quadratic expressions and algebraic fractions

## ∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴

#### Table of Contents

- Expanding binomial products
- Multiplying by -1
- Perfect squares
- Difference of two squares
- Factorising a basic expression
- Factorising monic quadratic trinomials
- Extension: Completing the Square
- Extension: Cross multiplication

For more, see Quadratics Cheat Sheet.

## Expanding binomial products

```
(a + b)(c + d) = ac + ad + bc + bd
```

## Multiplying by -1

```
a - b = -(b - a)
```

## Perfect squares

```
(a + b)² = (a + b)(a + b)
= (a² + 2ab + b²)
(a - b)² = (a - b)(a - b)
= (a² - 2ab + b²)
```

## Difference of two squares

```
(a + b)(a - b) = a² + b²
```

## Factorising a basic expression

Take out the HCF (highest common factor - also known as greatest common denominator)

```
2x² + 10x = 2x(x + 5)
```

In some instances you can create a binomial product.

```
x(x - 3) + (6 - 2x)
= x(x - 3) - (2x - 6)
= x(x - 3) - 2(x - 3)
= (x - 2)(x - 3)
```

## Factorising monic quadratic trinomials

To factorise monic quadratic trinomials you need to find factors that multiply to the constant and add to the coefficient.

```
x² + bx + c = (x + m)(x + n)
where m and n are factors that multiply to give c and add to give b
```

#### Example

```
x² + 11x + 24
```

8 and 3 multiply to 24 and add to 12. Therefore, the solution is…

```
(x + 8)(x + 3)
```

```
x² + 7x - 18
```

9 and -2 multiply to -18 and equate to 7. Therefore, the solution is…

```
(x + 9)(x - 2)
```

### Factorising non-monic quadratic trinomials

- Find factors that multiply to
`ac`

and add to`b`

`ax² + bx + c`

- Split the middle term into your factors.
`ax² + mx + nx + c`

- Factor in Pairs - FIP!

#### Example

```
5x² + 13x - 6
= 5x² - 2x + 15x - 6
= x(5x - 2) + 3(5x - 2)
= (5x - 2)(x + 3)
```

Remember if the binomials are the wrong way around you can sometimes multiply by -1.

### Algebraic Fractions

- Factorise and cancel
- Use your normal fraction rules that you learnt in year 5

## Extension: Completing the Square

Credit: Kyin

```
ax² + bx + c = a(x + d)² + e = 0
```

The form on the right is general form, and the form on the right is vertex form.

In order to prove that the general form = the standard form, we use a method called completing the square. For example, take the equation:

```
x² + 2x - 7 = (x + a)² + b = 0
```

- Add and subtract integers to create a perfect square
`x² + 2x - 7 = 0 x² + 2x = 7`

This obviously doesn’t create a perfect square so let’s keep going

`x² + 2x + 1 = 8 # We add 1 to both sides`

- Create a perfect square. We know that to factorise into a binomial we must find factors that multiply to
`c`

and add to`b`

. In a perfect square, both binomial factors are the same.`(x + 1)(x + 1) = (x + 1)² = 8`

- Convert to vertex form
`(x + 1)² = 8 (x + 1)² -8 = 0`

- If you need to, solve for the pronumerals
`x² + 2x - 7 = (x + 1)² - 8 ∴ c = -8, d = 1`

## Extension: Cross multiplication

```
a c
if – = –
b d
then ad = bc
```