Quadratic equations: Splitting the middle term (factoring quadratic equations)

A quadratic equation is also known as a trinomial. It has three terms in it: One is the term with `x^2`, the other is the term with `x` in it, and the third term is a constant number.

There are three parts to the method of splitting the middle term:

• Rearrange the given quadratic equation into the standard form (if not already in the standard form)
• Split the middle term
• Factor the quadratic equation.

Rearrange the given quadratic equation into the standard form (if not already in the standard form):

You need to first check if the given quadratic equation is in the standard form. A quadratic equation in the standard form looks like this:

`ax^2 + bx + c = 0`

For example, the following quadratic equations are in the standard form:

`2x^2 + 3x + 6 = 0`, `x^2 - 4x - 5 = 0`, and - `3x^2 - 6x + 5 = 0`

Now if a quadratic equation is not given in the standard form, then we will write it in the standard form. Like this:
Convert this quadratic equation into standard form:

`2x^2 = 4x + 6`

The steps will be like this:

Step	Explanation
2x 2 - 4x = 6	Subtracted 4x from both sides
2x 2 - 4x - 6 = 0	Subtracted 6 from both sides

Therefore the equation in standard form is:

`2x^2 - 4x - 6 = 0`

Similarly you will rewrite all quadratic equations in to the standard form in order to split the middle term and factorize.

Split the middle term:

The middle term of a quadratic equation is the term that has an 'x' in it but not the one that has 'x ²' in it. For example, the middle term of the following quadratic equation

`2x^2 - 4x - 6= 0`

is `-4x`. Remember to take the sign at the left of the middle term as well.
Now to split the middle term, you have to find two numbers (negative or positive) that add up to give the number in the middle term. For example, the following pairs of numbers add up to give the number -4 of the middle term:

• -2 + -2 = -4
• -3 + -1 = -4
• -5 + 1 = -4
• -6 + 2 = -4

Which of the above pairs of numbers should you take? There is another rule to determine that:

The product of the two parts of the middle term should be equal to the product of the first and last terms in the quadratic equation/expression.

The first and last terms in the quadratic equation

`2x^2 - 4x - 6 = 0`

are `2x^2` and `-6`. Their product is

`2x^2 * -6 = -12x^2`

So if you split the number of the middle term into two parts ' a' and ' b', then their product should be -12. Which of the pairs of numbers mentioned above multiply to give -12?

-6 * 2 = -12

There fore these two numbers -6 and 2 will form the correct two parts of the middle term -4x of the quadratic equation 2x ² - 4x - 6 = 0 .
Since -4x = -6x + 2x,
we can write -6x + 2x in place of -4x in the quadratic equation. That is,

2x ² - 4x - 6 = 0

is same as

2x ² - 6x + 2x - 6 = 0

Similarly you will split the middle term of all other quadratic equations.

Factorize:

After splitting the middle term of a quadratic equation, it becomes a four termed polynomial. That is, now it
In the equation

2x ² - 6x + 2x - 6 = 0

Making groups of the first two and the last two terms,

(2x ² - 6x) + (2x - 6) = 0

Now take a common factor from each group: from the first group we take 2x as the common factor and from the second group we take 2 as the common factor:

2x(x - 3) + 2(x - 3) = 0

Now you will always notice that there is a common factor in both groups. Here it is (x - 3) (Notice that (x - 3) is common in both the groups).
So take (x - 3) as the single common factor for both the groups and take it out as follows:

(x - 3)(2x + 2) = 0

Now, to completely factorize the equation, you can take 2 as the common factor from the second parenthesis,

2(x - 3)(x + 1) = 0

Therefore we factorized the quadratic equation 2x ² - 4x - 6 = 0 into (x - 3)(2x + 2) by the method of splitting the middle term

Interestingly, the equation you have obtained above by factorizing the quadratic equation is said to be in the intercept form.