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

## ∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴

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)
``````

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)
``````

1. Find factors that multiply to `ac` and add to `b`
``````ax² + bx + c
``````
2. Split the middle term into your factors.
``````ax² + mx + nx + c
``````
3. 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
``````
1. 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
``````
2. 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
``````
3. Convert to vertex form
``````(x + 1)² = 8
(x + 1)² -8 = 0
``````
4. 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