# Splitting the Middle Term

We now learn how to split the middle term.
Splitting the middle term is a method for factoring quadratic equations.

By the end of this section we'll know how to write quadratics in factored form: $ax^2+bx+c = \begin{pmatrix} mx + d \end{pmatrix} \begin{pmatrix} nx + e \end{pmatrix}$ For example, we'll know how to show that: $2x^2+7x+3 = \begin{pmatrix} 2x + 1 \end{pmatrix} \begin{pmatrix} x + 3 \end{pmatrix}$ We start by watching a tutorial to learn a five-step method for factornig quadratics. We then read through the five steps

## Tutorial

In the following tutorial we learn how to factor quadratics by splitting the middle term. Watch it now.

## Splitting the Middle Term

Given a quadratic: $ax^2+bx+c$ We'll often be required to factor it. We say we write it in factored form.

• Step 3: Split the Middle Term.
In step 2 we found $$p$$ and $$q$$ such that $$p+q = \text{coefficient of the middle term}$$, so: $p+q = b$ and consequently: $bx = px + qx$ and we can split the middle term to write: $ax^2 + bx + c = ax^2 + px + qx + c$

## Exercise 1

Write each of the following quadratics in factored form:

1. $$2x^2 + 7x + 3$$
2. $$6x^2+x-2$$
3. $$3x^2 - 13x + 4$$
4. $$3x^2-19x + 20$$
5. $$4x^2+9x+5$$

1. $$2x^2+7x + 3 = \begin{pmatrix} 2x + 1 \end{pmatrix} \begin{pmatrix} x + 3 \end{pmatrix}$$
2. $$6x^2 + x - 2 = \begin{pmatrix} 2x - 1 \end{pmatrix} \begin{pmatrix} 3x + 2 \end{pmatrix}$$
3. $$3x^2 - 13x + 4 = \begin{pmatrix}x - 4 \end{pmatrix} \begin{pmatrix} 3x - 1 \end{pmatrix}$$
4. $$3x^2 - 19 x + 20 = \begin{pmatrix} x - 5 \end{pmatrix} \begin{pmatrix} 3x - 4 \end{pmatrix}$$
5. $$4x^2 + 9x + 5 = \begin{pmatrix} 4x + 5 \end{pmatrix} \begin{pmatrix} x + 1 \end{pmatrix}$$

Now that we have seen how to factor quadratics, by splitting the middle term,

## Exercise 2

Solve each of the following quadratic equations by factoring (splitting the middle term):

1. $$4x^2 + 7x + 3 = 0$$
2. $$x^2+3x - 4 = 0$$
3. $$6x^2 - 7x +2 = 0$$
4. $$-2x^2 + x + 3 = 0$$