Factoring a quadratic equation without a middle term

This post is about factoring a quadratic equation without the middle term. In order to read the post about factoring a quadratic equation by splitting the middle term, go here:

Factoring a quadratic equation by splitting the middle term 

In order to read the post on factoring a quadratic equation by the box method, go here:

Factoring by box method 

A quadratic equation without b, or the middle term, is known as a pure quadratic equation. Examples of such equations are:

• `x^2 - 9 = 0`
• `x^2 + 23 = 27`
• `x^2 = 100` 

Notice that the above quadratic equations don't have 'b', or the x containing term, - they only have the x squared and a constant.

Factoring quadratic equations without a middle term (pure quadratic equations)

Method 1 

It is very simple to factor a quadratic equation without a middle term (a pure quadratic equation).

For example, we will factor the equation

`x^2 - 9 = 0`

Step 1: Move the number -9 to the other side. To do this, we will add 9 to both sides (since it is negative on the LHS)

`x^2 = 9`

Step 2: Now take square root of both sides, so we obtain

`x = +/- sqrt(9)`

Hence the answer is `x = +3` or `x = -3`, because both +3 and -3 are square roots of 9.
(learn more about why there are two square roots of a positive number)

Thus it is only a matter of finding the square root when you have to solve a quadratic equation without the middle term.

Method 2

The second method involved algebra. It is based on the algebraic difference between two squares. The particular formula used here is

`a^2 - b^2 = (a + b)(a - b)`

We will consider a pure quadratic equation as a difference of two squares. For example, the equation

`x^2 - 9 = 0`

can be written as a difference of two squares as follows:

`x^2 - 3^2 = 0`

Now applying the property `a^2 - b^2 = (a + b)(a - b)`, we get

`(x + 3)(x - 3)  = 0`

Now applying the zero product rule, we get

Either x = -3 or x = 3

Thus a quadratic equation/function that does not have a pure quadratic equations can be solved by two methods. The method 1 described above will be able to quickly solve all such quadratic equations.