To solve the quadratic equation [tex]\( x^2 + 4 = 8x + 5 \)[/tex], let's follow a detailed step-by-step process.
1. Rearrange the equation:
Start by moving all terms to one side of the equation to set it equal to zero.
[tex]\[
x^2 + 4 - 8x - 5 = 0
\][/tex]
Simplify the equation:
[tex]\[
x^2 - 8x - 1 = 0
\][/tex]
2. Identify the coefficients:
For a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
a = 1, \quad b = -8, \quad c = -1
\][/tex]
3. Apply the quadratic formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the formula:
[tex]\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1}
\][/tex]
Simplify the terms inside the square root and the expression:
[tex]\[
x = \frac{8 \pm \sqrt{64 + 4}}{2}
\][/tex]
Further simplifying:
[tex]\[
x = \frac{8 \pm \sqrt{68}}{2}
\][/tex]
Since [tex]\(\sqrt{68} = \sqrt{4 \cdot 17} = 2\sqrt{17}\)[/tex], we have:
[tex]\[
x = \frac{8 \pm 2\sqrt{17}}{2}
\][/tex]
Simplify the expression:
[tex]\[
x = 4 \pm \sqrt{17}
\][/tex]
So, the solutions to the quadratic equation [tex]\( x^2 + 4 = 8x + 5 \)[/tex] are [tex]\( x = 4 - \sqrt{17} \)[/tex] and [tex]\( x = 4 + \sqrt{17} \)[/tex].
The correct answer is:
D. [tex]\( x = 4 \pm \sqrt{17} \)[/tex]
