To solve the quadratic equation [tex]\( x^2 + 13 = 8x + 37 \)[/tex], we need to follow these steps:
1. Move all terms to one side of the equation to set it to zero:
[tex]\[
x^2 + 13 - 8x - 37 = 0
\][/tex]
Simplify the expression:
[tex]\[
x^2 - 8x - 24 = 0
\][/tex]
2. Solve the quadratic equation [tex]\( x^2 - 8x - 24 = 0 \)[/tex]. To do this, we can use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = -24 \)[/tex].
3. Plug in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the quadratic formula:
[tex]\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot (-24)}}{2 \cdot 1}
\][/tex]
Simplify inside the square root:
[tex]\[
x = \frac{8 \pm \sqrt{64 + 96}}{2}
\][/tex]
Further simplification gives us:
[tex]\[
x = \frac{8 \pm \sqrt{160}}{2}
\][/tex]
4. Simplify the square root of 160:
[tex]\[
\sqrt{160} = \sqrt{16 \cdot 10} = 4\sqrt{10}
\][/tex]
So the expression now becomes:
[tex]\[
x = \frac{8 \pm 4\sqrt{10}}{2}
\][/tex]
5. Finally, divide both the terms in the numerator by 2:
[tex]\[
x = 4 \pm 2\sqrt{10}
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\( x^2 + 13 = 8x + 37 \)[/tex] are:
[tex]\[
x = 4 \pm 2\sqrt{10}
\][/tex]
The correct answer is:
C. [tex]\( x = 4 \pm 2\sqrt{10} \)[/tex]
