Certainly! Let's go through the steps to solve the given equation [tex]\( x^2 - 10x + 10 = 0 \)[/tex] by completing the square and then finding its solutions.
### Step 1: Completing the Square
To complete the square, we follow these steps:
1. Start with the quadratic equation:
[tex]\[
x^2 - 10x + 10 = 0
\][/tex]
2. Move the constant term to the other side of the equation:
[tex]\[
x^2 - 10x = -10
\][/tex]
3. To complete the square, we take half of the coefficient of the [tex]\( x \)[/tex] term, square it, and add it to both sides. The coefficient of [tex]\( x \)[/tex] is [tex]\(-10\)[/tex], so half of this is [tex]\(-5\)[/tex], and squaring it gives us [tex]\(25\)[/tex]. Add and subtract [tex]\(25\)[/tex] on the left side:
[tex]\[
x^2 - 10x + 25 - 25 = -10
\][/tex]
Simplifies to:
[tex]\[
x^2 - 10x + 25 = 15
\][/tex]
4. Now the left side is a perfect square trinomial:
[tex]\[
(x - 5)^2 = 15
\][/tex]
Therefore, the equation in its completed square form is:
[tex]\[
(x - 5)^2 = 15
\][/tex]
### Step 2: Solving the Completed Square Equation
Next, we solve for [tex]\( x \)[/tex].
1. Take the square root of both sides:
[tex]\[
x - 5 = \pm \sqrt{15}
\][/tex]
2. Solve for [tex]\( x \)[/tex] by isolating it:
[tex]\[
x = 5 \pm \sqrt{15}
\][/tex]
Thus, the solutions to the equation are:
[tex]\[
x = 5 - \sqrt{15} \quad \text{and} \quad x = 5 + \sqrt{15}
\][/tex]
### Conclusion
The solutions to the equation [tex]\( x^2 - 10x + 10 = 0 \)[/tex] are [tex]\(\boxed{x = 5 - \sqrt{15}} \)[/tex] and [tex]\(\boxed{x = 5 + \sqrt{15}} \)[/tex].
Therefore, the correct answer is:
(B) [tex]\( x = 5 \pm \sqrt{15} \)[/tex]
