Let's solve the equation [tex]\( x^2 - 10x - 1 = 13 \)[/tex] by completing the square. Here is the step-by-step solution:
1. Start with the given equation:
[tex]\[
x^2 - 10x - 1 = 13
\][/tex]
2. Move the constant term on the left side of the equation to the right side:
[tex]\[
x^2 - 10x = 13 + 1
\][/tex]
[tex]\[
x^2 - 10x = 14
\][/tex]
3. Complete the square:
To complete the square, we need to add and subtract the square of half the coefficient of [tex]\( x \)[/tex]. The coefficient of [tex]\( x \)[/tex] is [tex]\(-10\)[/tex], so half of it is [tex]\(-5\)[/tex] and squaring it gives 25.
[tex]\[
x^2 - 10x + 25 - 25 = 14
\][/tex]
[tex]\[
(x - 5)^2 - 25 = 14
\][/tex]
4. Simplify the equation:
[tex]\[
(x - 5)^2 - 25 = 14
\][/tex]
[tex]\[
(x - 5)^2 = 14 + 25
\][/tex]
[tex]\[
(x - 5)^2 = 39
\][/tex]
5. Take the square root of both sides:
[tex]\[
x - 5 = \pm \sqrt{39}
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
[tex]\[
x - 5 = \sqrt{39}
\][/tex]
[tex]\[
x = 5 + \sqrt{39}
\][/tex]
and
[tex]\[
x - 5 = -\sqrt{39}
\][/tex]
[tex]\[
x = 5 - \sqrt{39}
\][/tex]
So, the solutions to the equation are [tex]\( x = 5 + \sqrt{39} \)[/tex] and [tex]\( x = 5 - \sqrt{39} \)[/tex].
Checking the provided options:
A. [tex]\( 5 - \sqrt{39} \)[/tex] — Correct
B. [tex]\( -10 - \sqrt{24} \)[/tex] — Incorrect
C. [tex]\( 5 + \sqrt{39} \)[/tex] — Correct
D. [tex]\( 10 + \sqrt{24} \)[/tex] — Incorrect
Thus, the correct choices are:
- [tex]\( 5 - \sqrt{39} \)[/tex]
- [tex]\( 5 + \sqrt{39} \)[/tex]
Hence, the correct answers are:
A. [tex]\( 5 - \sqrt{39} \)[/tex]
C. [tex]\( 5 + \sqrt{39} \)[/tex]
