Here are a few quadratic expressions you can use for practising this method.
- \(4x^2+7x+3\)
We can partially fill in a multiplication grid with this quadratic:
| \(\ldots x\) | \(\ldots\) | |
|---|---|---|
| \(\ldots x\) | \(4x^2\) | |
| \(\ldots\) | \(+3\) |
The remaining two cells must add to \(+7x\) and multiply to \(12x^2\), so they are \(+4x\) and \(+3x\), giving:
| \(\ldots x\) | \(\ldots\) | |
|---|---|---|
| \(\ldots x\) | \(4x^2\) | \(+4x\) |
| \(\ldots\) | \(+3x\) | \(+3\) |
We fill in the outside of the grid by looking for the highest common factor of the first column, which in this case is \(x\).
| \(x\) | \(\ldots\) | |
|---|---|---|
| \(\ldots x\) | \(4x^2\) | \(+4x\) |
| \(\ldots\) | \(+3x\) | \(+3\) |
We then fill in the rest of the grid to make the multiplication work.
- \(3x^2+13x-10\)
| \(\ldots x\) | \(\ldots\) | |
|---|---|---|
| \(\ldots x\) | \(3x^2\) | |
| \(\ldots\) | \(-10\) |
The remaining two cells must add to \(+13x\) and multiply to \(-30x^2\), so they are \(-2x\) and \(+15x\), giving:
| \(\ldots x\) | \(\ldots\) | |
|---|---|---|
| \(\ldots x\) | \(3x^2\) | \(-2x\) |
| \(\ldots\) | \(+15x\) | \(-10\) |
We fill in the outside of the grid by looking for the highest common factor of the first column and then fill in the rest of the grid to make the multiplication work.
- \(6x^2-5x+1\)
We can partially fill in a multiplication grid with this quadratic:
| \(\ldots x\) | \(\ldots\) | |
|---|---|---|
| \(\ldots x\) | \(6x^2\) | |
| \(\ldots\) | \(+1\) |
The remaining two cells must add to \(-5x\) and multiply to \(6x^2\), so they are \(-2x\) and \(-3x\), giving:
| \(\ldots x\) | \(\ldots\) | |
|---|---|---|
| \(\ldots x\) | \(6x^2\) | \(-2x\) |
| \(\ldots\) | \(-3x\) | \(+1\) |
We fill in the outside of the grid by looking for the highest common factor of the first column and then fill in the rest of the grid to make the multiplication work.
- \(4x^2-2x-12\)
| \(\ldots x\) | \(\ldots\) | |
|---|---|---|
| \(\ldots x\) | \(4x^2\) | |
| \(\ldots\) | \(-12\) |
The remaining two cells must add to \(-2x\) and multiply to \(-48x^2\), so they are \(+6x\) and \(-8x\), giving:
| \(\ldots x\) | \(\ldots\) | |
|---|---|---|
| \(\ldots x\) | \(4x^2\) | \(+6x\) |
| \(\ldots\) | \(-8x\) | \(-12\) |
Why are there two different possibilities here?
- \(2x^2-3xy-2y^2\)
This looks a little different, as there are now appearances of \(x\) and \(y\). But we can still partially fill in a multiplication grid, only this time, the factors are going to involve \(x\) and \(y\) which means the grid will have this form:
| \(\ldots x\) | \(\ldots y\) | |
|---|---|---|
| \(\ldots x\) | \(2x^2\) | |
| \(\ldots y\) | \(-2y^2\) |
The remaining two cells must add to \(-3xy\) and multiply to \(-4x^2y^2\).
Pairs of \(xy\) terms which multiply to \(-4x^2y^2\) are \(4xy\) and \(-xy\), \(-4xy\) and \(xy\), \(2xy\) and \(-2xy\). Only \(-4xy\) and \(+xy\) sum to \(-3xy\). This gives the grid:
| \(\ldots x\) | \(\ldots y\) | |
|---|---|---|
| \(\ldots x\) | \(2x^2\) | \(-4xy\) |
| \(\ldots y\) | \(+xy\) | \(-2y^2\) |
Reflecting
How does this method of factorising quadratics compare to other methods you know?
You may have learnt some other methods, and using whichever seems most appropriate to the context is often sensible.
