To solve the equation [tex]\(x^2 - 10x - 7 = 0\)[/tex] by completing the square, follow these steps:
1. Start with the original equation:
[tex]\[
x^2 - 10x - 7 = 0
\][/tex]
2. Move the constant term to the other side of the equation by adding 7 to both sides:
[tex]\[
x^2 - 10x = 7
\][/tex]
3. To complete the square, take half of the coefficient of [tex]\(x\)[/tex], square it, and add it to both sides of the equation. The coefficient of [tex]\(x\)[/tex] is [tex]\(-10\)[/tex]. Half of [tex]\(-10\)[/tex] is [tex]\(-5\)[/tex], and squaring it gives [tex]\(25\)[/tex]:
[tex]\[
x^2 - 10x + 25 = 7 + 25
\][/tex]
4. Rewrite the left side as a perfect square and simplify the right side:
[tex]\[
(x - 5)^2 = 32
\][/tex]
5. Take the square root of both sides:
[tex]\[
x - 5 = \pm \sqrt{32}
\][/tex]
6. Solve for [tex]\(x\)[/tex] by isolating it:
[tex]\[
x = 5 \pm \sqrt{32}
\][/tex]
Therefore, the solution for the equation [tex]\(x^2 - 10x - 7 = 0\)[/tex] is:
- [tex]\(x = 5 \pm \sqrt{32}\)[/tex]
So, in Input Box 1, you should enter the constant [tex]\(5\)[/tex], and in Input Box 2, you should enter the number inside the radical, which is [tex]\(32\)[/tex].
[tex]\[
x = \boxed{5} \pm \sqrt{ \boxed{32}}
\][/tex]
