Splitting the middle term

Introduction to splitting the middle term:

Splitting the middle term of a quadratic equation is a method used in factoring of quadratic equations. Splitting the middle term means that you have to rewrite the middle term of a quadratic expression as a sum of two terms. This means that in splitting the middle term, you have split the middle term into two parts.
Let us learn splitting the middle term with the help of an example:

Example of splitting the middle term:

For example, consider the following quadratic equation:

x² + 3x - 4

For splitting the middle term, you will take the middle term of the above quadratic: 3x, and split it into two parts like:

3x = 4x - x

Now, 3x can be written as (4x - x), so the quadratic expression can be written as

x² + 4x - x - 4

Thus, by splitting the middle term, we have converted a three termed quadratic expression to a four termed quadratic expression. Now, you have to factorize the above four termed quadratic expression. In order to do that, make two groups - the first group consisting of the first two terms x² and 4x and the second group consisting of the next two terms -x and -4. Thus, you get

(x² + 4x) - (x + 4)

Now, from each group, take out the highest common factor. The highest common factor of the first group is x and that of the second group is 1. Thus the equation becomes

x(x + 4) - 1(x + 4)

Now, if you have done the splitting the middle term correctly, then you should get one expression common in both the groups. Here (x + 4) is the expression common in both the groups. So you can take it as the common factor, and again rewrite the equation as

(x + 4)(x - 1)

Thus, by the method of splitting the middle term, we factored the quadratic expression x² + 3x - 4 into (x + 4)(x - 1)

Conclusion for splitting the middle term

Thus, by the method of splitting the middle term, you convert a quadratic expression into an algebraic expression that is a product of two simpler (linear) algebraic expressions.