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### 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.

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.

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.