Sure, let's solve the equation step-by-step and then validate each option given.
We start with the original equation:
[tex]\[ x^2 - 10x + 25 = 17 \][/tex]
First, we subtract 17 from both sides to simplify it to a standard form:
[tex]\[ x^2 - 10x + 8 = 0 \][/tex]
Next, we solve this quadratic equation using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a=1 \)[/tex], [tex]\( b=-10 \)[/tex], and [tex]\( c=8 \)[/tex].
1. Calculate the discriminant:
[tex]\[ b^2 - 4ac = (-10)^2 - 4(1)(8) = 100 - 32 = 68 \][/tex]
2. Find the square root of the discriminant:
[tex]\[ \sqrt{68} = \sqrt{4 \times 17} = 2\sqrt{17} \][/tex]
3. Apply the quadratic formula:
[tex]\[ x = \frac{-(-10) \pm 2\sqrt{17}}{2(1)} = \frac{10 \pm 2\sqrt{17}}{2} \][/tex]
4. Simplify the expression:
[tex]\[ x = \frac{10 + 2\sqrt{17}}{2} = 5 + \sqrt{17} \][/tex]
[tex]\[ x = \frac{10 - 2\sqrt{17}}{2} = 5 - \sqrt{17} \][/tex]
So, the solutions to the equation are:
[tex]\[ x = 5 + \sqrt{17} \][/tex]
[tex]\[ x = 5 - \sqrt{17} \][/tex]
Next, we check each given option:
- A. [tex]\( x = -\sqrt{17} + 5 \)[/tex]:
- Comparing [tex]\(-\sqrt{17} + 5\)[/tex] with [tex]\(5 \pm \sqrt{17}\)[/tex], there is no match.
- B. [tex]\( x = -\sqrt{17} - 5 \)[/tex]:
- Comparing [tex]\(-\sqrt{17} - 5\)[/tex] with [tex]\(5 \pm \sqrt{17}\)[/tex], there is no match.
- C. [tex]\( x = \sqrt{17} + 5 \)[/tex]:
- Comparing [tex]\(\sqrt{17} + 5\)[/tex] with [tex]\(5 + \sqrt{17}\)[/tex], it matches.
- D. [tex]\( x = \sqrt{8} + 5 \)[/tex]:
- Comparing [tex]\(\sqrt{8} + 5\)[/tex] with [tex]\(5 \pm \sqrt{17}\)[/tex], there is no match.
- E. [tex]\( x = \sqrt{17} - 5 \)[/tex]:
- Comparing [tex]\(\sqrt{17} - 5\)[/tex] with [tex]\(5 - \sqrt{17}\)[/tex], there is no match.
- F. [tex]\( x = -\sqrt{8} - 5 \)[/tex]:
- Comparing [tex]\(-\sqrt{8} - 5\)[/tex] with [tex]\(5 \pm \sqrt{17}\)[/tex], there is no match.
So, none of the given options A, B, D, E, or F match our solutions [tex]\( 5 + \sqrt{17} \)[/tex] and [tex]\( 5 - \sqrt{17} \)[/tex]. The valid solution from the given options is:
- C. [tex]\( x = \sqrt{17} + 5 \)[/tex]
Hence, the correct solutions are:
[tex]\[ \boxed{C} \][/tex]
