Solving quadratic equations by splitting the middle term

Introduction

Quadratic equations can only be solved by the method of splitting the middle term when they are written in the standard form. A quadratic equation is said to be in its standard form when the left hand side expression is a quadratic expression in the standard or general form, and is equated with zero on the right
hand side.

The left hand side expression of such a quadratic equation is factored into a product of two linear algebraic expressions by the method of splitting its middle term. Then, by applying the Zero Product Rule the roots of
the quadratic equation are obtained.

Thus to solve any quadratic equation by splitting the middle term, these general steps are followed:

• Convert the quadratic equation to the standard form
• Factor the left hand side (ie, LHS) by the method of splitting the middle term
• Obtain the roots by applying the Zero Product Rule

Understanding with an example

Let us take an example to understand it clearly,

Solve:    x² + 4x = -4

Solution:

Step 1: Convert the given equation to its standard form

 x² + 4x + 4 = 0

Step 2: Factor the LHS by the method of splitting the middle term

The middle term is 4x and the master product is 4x². Thus, the two parts are 2x and 2x, because 2x times 2x is 4x² and 2x + 2x equals 4x. Thus, the LHS expression becomes,

x² + 2x + 2x + 4

• Form two groups: (x² + 2x) + (2x + 4)
• Take common factors from each group: x(x + 2) + 2(x + 2)
• Take common factor from both groups: (x + 2)(x + 2)

Thus the equation becomes (x + 2)(x + 2) = 0, in which the LHS expression is completely factored.

Step 3: Obtain the roots by applying the Zero Product Rule

By the Zero Product Rule, either (x + 2) = 0 or (x + 2) = 0, and in both, x = -2.

Hence the roots of the given quadratic equation are -2 and -2.