Complete the square to solve the equation below. Check all that apply.

[tex]\[ x^2 - 10x - 1 = 13 \][/tex]

A. [tex]\( 5 - \sqrt{39} \)[/tex]

B. [tex]\( -10 - \sqrt{24} \)[/tex]

C. [tex]\( 5 + \sqrt{39} \)[/tex]

D. [tex]\( 10 + \sqrt{24} \)[/tex]



Answer :

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]