Let's solve the equation [tex]\( x^2 - 10x - 4 = 10 \)[/tex] by completing the square.
1. Start with the given equation:
[tex]\[
x^2 - 10x - 4 = 10
\][/tex]
2. Move the constant term (-4) to the right side of the equation:
[tex]\[
x^2 - 10x = 10 + 4
\][/tex]
[tex]\[
x^2 - 10x = 14
\][/tex]
3. To complete the square, add and subtract the square of half the coefficient of [tex]\( x \)[/tex] (which is [tex]\( -10 \)[/tex]):
[tex]\[
x^2 - 10x + \left(\frac{-10}{2}\right)^2 = 14 + \left(\frac{-10}{2}\right)^2
\][/tex]
[tex]\[
x^2 - 10x + 25 = 14 + 25
\][/tex]
[tex]\[
x^2 - 10x + 25 = 39
\][/tex]
4. Rewrite the left side as a perfect square:
[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]\[
x - 5 = \sqrt{39} \quad \text{or} \quad x - 5 = -\sqrt{39}
\][/tex]
[tex]\[
x = 5 + \sqrt{39} \quad \text{or} \quad x = 5 - \sqrt{39}
\][/tex]
Therefore, the solutions to the equation [tex]\( x^2 - 10x - 4 = 10 \)[/tex] are:
[tex]\( x = 5 + \sqrt{39} \)[/tex] and [tex]\( x = 5 - \sqrt{39} \)[/tex].
Among the options provided:
- A. [tex]\( 10 + \sqrt{24} \)[/tex]
- B. [tex]\( 10 \cdot \sqrt{24} \)[/tex]
- C. [tex]\( 5 - \sqrt{39} \)[/tex]
- D. [tex]\( 5 + \sqrt{39} \)[/tex]
The correct answers are:
- C. [tex]\( 5 - \sqrt{39} \)[/tex]
- D. [tex]\( 5 + \sqrt{39} \)[/tex]
