On this page, we will show you how to factor 25x squared plus 81 (25x^2+81) using the sum of squares formula. In addition to factoring 25x^2+81, we will also verify that our answer is correct by calculating the product of the factors we found.

Before we begin, note that 25 and 81 are both perfect squares. Adding one perfect square to another is called sum of squares or sum of two squares.

Furthermore, you cannot factor sum of squares with real numbers. However, you can solve 25x^2+81 with a complex or imaginary number. We will make our imaginary number (i) equal to -1.

The sum of squares formula we will use to factor 25x^2+81 with our imaginary number is as follows:

a

^{2}+ b

^{2}= (a + bi) • (a − bi)

We start by setting up our problem in mathematical terms like this:

25x

^{2}+ 81

As you can see, our problem does not look like the left side of the formula, so first we have to convert our problem to match the left side like so:

25x

^{2}+ 81 = (5x)

^{2}+ 9

^{2}

Now that it matches, we can simply plug in 5x for a and 9 for b into the formula to get the factors of 25x^2+81:

**(5x + 9i) • (5x − 9i)**

To verify that our answer is correct, we can calculate the product of the factors to see if it equals 25x^2+81. And it does, as illustrated below:

= (5x + 9i) • (5x − 9i)

= 5x(5x - 9i) + 5x(5x - 9i)

= 25x

^{2}- 45xi + 45xi - 81i

^{2}

- 45xi + 45xi evens out and -i

^{2}equals -1, so now we get this:

= 25x

^{2}- 81(-1)

= 25x

^{2}+ 81

**Factoring Sum of Squares**

Do you need to factor another polynomial using the sum of squares formula? If so, enter it below, but remember that both numbers must be perfect squares.

**Factor 25x^2+100**

We hope this step-by-step tutorial to teach you how to factor 25x squared plus 81 was helpful. Do you want to learn more? Go here for the next tutorial on our list.

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