Factorise quadratics

We have looked at factorising quadratic where the coeffect of the $x^2$ term is 1. If we have $5x^2$, we will need to follow a slightly different method to factorise.

1. Write the quadratic in standard form:

Ensure the quadratic is written as $a{x}^2+bx+c$.

2. Identify $a$, $b$, and $c$:

Recognise the values of $a$ (the coefficient of $x^2$), $b$ (the coefficient of $x$), and $c$ (the constant term).

3. Find two numbers that multiply to $ac$ and add to $b$:

Multiply $a$ and $c$. Find two numbers that multiply to this $ac$ value and add up to $b$. These two numbers will help split the middle term.

4. Rewrite the quadratic by splitting the middle term:

Replace $bx$ with two terms using the numbers found in the previous step. This splits the quadratic into four terms. For example, if the quadratic is $6x^2+7x+2$, find two numbers that multiply to $6\times 2=12$ and add to $7$, which are $3$ and $4$. Rewrite $6x^2+7x+2$ as $6x^2+3x+4x+2$.

5. Factorise by grouping:

Group the terms into two pairs: $(6x^2+3x)+(4x+2)$. Factor out the common factor from each pair: $3x(2x+1)+2(2x+1)$.

6. Factorise out the common binomial:

If both groups have a common factor (a binomial), factor it out. For the example, factor out $(2x+1)$: $(2x+1)(3x+2)$.

7. Check the factorisation:

Expand the factors to confirm they multiply to the original quadratic.

Example

Factorise $6x^2+7x+2$.

1. Write in standard form:

$6x^2+7x+2$.

2. Identify $a=6$, $b=7$, $c=2$:

The values are $a=6$, $b=7$, and $c=2$.

3. Find two numbers that multiply to $6\times 2=12$ and add to $7$:

The numbers $3$ and $4$ work.

4. Rewrite the expression:

Rewrite $6x^2+7x+2$ as $6x^2+3x+4x+2$.

5. Group terms:

$(6x^2+3x)+(4x+2)$.

6. Factor out the common factor:

Factor out $3x(2x+1)+2(2x+1)$.

7. Factor out $(2x+1)$:

$(2x+1)(3x+2)$.

The factorised form is:

$(2x+1)(3x+2)$.