RadfordMathematics.com

Online Mathematics Book

Splitting the Middle Term

Factoring Quadratics


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. \(4x^2+9x+5\)

Answers Without Working

  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. \(4x^2 + 9x + 5 = \begin{pmatrix} 4x + 5 \end{pmatrix} \begin{pmatrix} x + 1 \end{pmatrix}\)

Solving Quadratic Equations

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. \(-2x^2 + x + 3 = 0\)

Solution Without Working

Solution With Working