Answer
The solutions are $x = 6, \text{ and } x=-6$.
Work Step by Step
Rewrite the equation in standard form:
$x^2 - 36 = 0$
We see that this equation can be factored using the formula to factor the difference of two squares. According to this formula:
$a^2 - b^2 = (a + b)(a - b)$
In the equation $x^2 - 36 = 0$, $a$ is the $\sqrt {x^2}$ or $x$, and $b$ is $\sqrt {36}$ or $6$. We plug these values into the formula:
$(x + 6)(x - 6) = 0$
The Zero-Product Property states that if the product of two factors equals $0$, then either factor equals $0$, or both factors equal $0$. Therefore, we can set each factor equal to $0$ to solve for $x$.
Let us set the first factor equal to $0$:
$x + 6 = 0$
$x = -6$
Let us set the second factor equal to $0$:
$x - 6 = 0$
$x = 6$
The solutions are $x = -6, 6$.
To check if the solution is correct, we plug in the values we got for $x$ into the original equation to see if both sides are equal. Let's plug in $6$ first:
$(6)^2 + 1 = 37$
$36 + 1 = 37$
$37 = 37$
This solution is correct.
Let us look at $-6$:
$(-6)^2 + 1 = 37$
$36 + 1 = 37$
$37 = 37$
This solution is also correct.
