On this page, we will show you how to factor x squared plus 49 (x^2+49)using the sum of squares formula. In addition to factoring x^2+49, we will also verify that our answer is correct bycalculating the product of the factors we found.

Before we begin, note that 1 and 49 are both perfect squares. Adding one perfect square to another is calledsum of squares or sum of two squares.

Furthermore, you cannot factor sum of squares with real numbers.However, you can solve x^2+49 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 x^2+49 with our imaginary number is as follows:

a2 + b2 = (a + bi) • (a − bi)

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

x2 + 49

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

x2 + 49 = x2 + 72

Now that it matches, we can simply plug in x for a and 7 for b into the formula to get the factors of x^2+49:

(x + 7i) • (x − 7i)

To verify that our answer is correct, we can calculate the product of the factors to see if it equals x^2+49. And it does, asillustrated below:

= (x + 7i) • (x − 7i)

= x(x - 7i) + x(x - 7i)

= x2 - 7xi + 7xi - 49i2

- 7xi + 7xi evens out and -i2 equals -1, so now we get this:

= x2 - 49(-1)

= x2 + 49

Factoring Sum of SquaresDo you need to factor another polynomial using the sum of squares formula? If so, enter it below, but rememberthat both numbers must be perfect squares.