Quadratics & Factorising

What is Factorising?

Factorising is the process of breaking down a quadratic equation into simpler expressions (factors) that can be multiplied together to give the original quadratic. It’s like taking apart a big Lego structure into smaller blocks.

The general quadratic equation: $$ax^2 + bx + c = 0$$

To factorise, you look for two expressions like $(x + p)$ and $(x + q)$ such that: $$(x +p)(x + q) = ax^2 + bx + c$$

 

When Can You Use Factorising?

You can solve a quadratic equation by factorising if it can be rewritten as a product of two brackets. This method works well for simple quadratics where $a = 1$, or quadratics that can be simplified.

 

Steps for Factorising

1. Write the quadratic equation in standard form: $$ax^2 + bx + c = 0$$

2. Find two numbers that:

   • Multiply to give $c$ (the constant term).
   • Add to give $b$ (the coefficient of $x$).

3. Split the middle term using the two numbers.

4. Group terms in pairs and factorise each pair.

5. Write the equation as two brackets.

6. Solve for $x$ by setting each bracket equal to 0.

 

Examples

Example 1

Solving $x^2 + 5x + 6 = 0$

1. Write in standard form: $$x^2 + 5x + 6 = 0$$
2. Find two numbers that multiply to $6$ and add to $5$:
   • Numbers are $2$ and $3$ because: $$2 \times 3 = 6,  2 + 3 = 5$$
3. Rewrite the quadratic: $$x^2 + 2x + 3x + 6 = 0$$
4. Group terms: $$(x^2 + 2x) + (3x + 6) = 0$$
5. Factorise each group: $$x(x + 2) + 3(x + 2) = 0$$
6. Write as two brackets: $$(x + 2)(x + 3)$$
7. Solve: $$x + 2 = 0 \rArr x = -2 \\ x + 3 = 0 \rArr x = -3$$

Solution: $$x = -2  \text{or}  x = -3$$

 

 

 

 

 

 

Steps for Factorising: Advanced (where $a \not = 1$ )

When the coefficient of the square i.e $a \not = 1$ the method used is very slightly different with one extra step. The method is:

1. Write the quadratic equation in standard form: $$ax^2 + bx + c = 0$$

2. First multiply the $a$ and $c$ terms together.

3. Now find two numbers that:

   • Multiply to give your answer to part 2 ($a \times c$).
   • Add to give $b$ (the coefficient of $x$).

4. Split the middle term using the two numbers.

5. Group terms in pairs and factorise each pair.

6. Write the equation as two brackets.

7. Solve for $x$ by setting each bracket equal to 0.

 

Example 2

Solving $2x^2 + 7x + 3 = 0$

1. Write in standard form: $2x^2 + 7x + 3 = 0$
2. Multiply $a$ and $c$:  $2 \times 3 = 6$
3. Find two numbers that multiply to $6$ and add to $7$:
   • Numbers are $6$ and $1$ because: $$6 \times 1 = 6,  6 + 1 = 7$$
4. Rewrite the middle term: $$2x^2 + 6x + x + 3 = 0$$
5. Group terms: $$(2x^2 + 6x) + (x + 3) = 0$$
6. Factorise each group: $$2x(x + 3) + 1(x + 3) = 0$$
7. Write as two brackets: $$(2x + 1)(x + 3) = 0$$
8. Solve: $$2x + 1 = 0 \rArr x = -\frac{1}{2} \\ x + 3 = 0 \rArr x = -3$$

Solution: $$x = -\frac{1}{2}  \text{or}  x = -3$$