To determine which of the provided options are solutions to the equation [tex]\(x^2 - 10x + 25 = 17\)[/tex], we'll follow these steps:
1. Simplify the equation:
[tex]\[
x^2 - 10x + 25 = 17
\][/tex]
Subtract 17 from both sides to set the equation to zero:
[tex]\[
x^2 - 10x + 25 - 17 = 0
\][/tex]
This simplifies to:
[tex]\[
x^2 - 10x + 8 = 0
\][/tex]
2. Factor or use the quadratic formula:
The quadratic formula for [tex]\(ax^2 + bx + c = 0\)[/tex] is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For the equation [tex]\(x^2 - 10x + 8 = 0\)[/tex]:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -10\)[/tex]
- [tex]\(c = 8\)[/tex]
Plug these values into the quadratic formula:
[tex]\[
x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}
\][/tex]
Simplify the expression under the square root:
[tex]\[
x = \frac{10 \pm \sqrt{100 - 32}}{2}
\][/tex]
[tex]\[
x = \frac{10 \pm \sqrt{68}}{2}
\][/tex]
[tex]\[
x = \frac{10 \pm 2\sqrt{17}}{2}
\][/tex]
Simplify the fractions:
[tex]\[
x = 5 \pm \sqrt{17}
\][/tex]
So, the solutions to the equation [tex]\(x^2 - 10x + 8 = 0\)[/tex] are:
[tex]\[
x = 5 + \sqrt{17} \quad \text{and} \quad x = 5 - \sqrt{17}
\][/tex]
3. Check the provided options:
- A. [tex]\(x = \sqrt{17} - 5\)[/tex]: This is not one of our solutions.
- B. [tex]\(x = \sqrt{8} + 5\)[/tex]: This is not one of our solutions.
- C. [tex]\(x = -\sqrt{8} - 5\)[/tex]: This is not one of our solutions.
- D. [tex]\(x = \sqrt{17} + 5\)[/tex]: This matches [tex]\(5 + \sqrt{17}\)[/tex].
- E. [tex]\(x = -\sqrt{17} - 5\)[/tex]: This is not one of our solutions.
- F. [tex]\(x = -\sqrt{17} + 5\)[/tex]: This matches [tex]\(5 - \sqrt{17}\)[/tex].
Based on this analysis, the solutions to the equation [tex]\(x^2 - 10x + 25 = 17\)[/tex] are:
- D. [tex]\(x = \sqrt{17} + 5\)[/tex]
- F. [tex]\(x = -\sqrt{17} + 5\)[/tex]
