Year 9 Maths Chapter 8 - Quadratic expressions and algebraic fractions

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Table of Contents

• Expanding binomial products
• Multiplying by -1
• Perfect squares
• Difference of two squares
• Factorising a basic expression
• Factorising monic quadratic trinomials
  • Example
  • Factorising non-monic quadratic trinomials
    • Example
  • Algebraic Fractions
• 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

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


then ad = bc