To solve the quadratic equation [tex]\(x^2 + 8x = 10\)[/tex] by completing the square, follow these steps:
1. Write the equation in a standard form suitable for completing the square:
[tex]\[
x^2 + 8x = 10
\][/tex]
2. Move the constant term to the other side of the equation:
[tex]\[
x^2 + 8x - 10 = 0
\][/tex]
3. To complete the square, focus on the quadratic and linear terms (ignore the constant for now). 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] here is 8:
- Half of 8 is 4.
- Squaring 4 gives us 16.
4. Add and subtract this square (16) inside the equation to maintain equality. Adding it on the other side compensates for maintaining balance:
[tex]\[
x^2 + 8x + 16 - 16 - 10 = 0
\][/tex]
5. Rewrite the left-hand side as a perfect square trinomial:
[tex]\[
(x + 4)^2 - 16 - 10 = 0
\][/tex]
6. Combine the constants on the other side:
[tex]\[
(x + 4)^2 - 26 = 0
\][/tex]
7. Move the constants term to the right side to isolate the perfect square:
[tex]\[
(x + 4)^2 = 26
\][/tex]
8. Take the square root of both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x + 4 = \pm\sqrt{26}
\][/tex]
9. Isolate [tex]\(x\)[/tex] by subtracting 4 from both sides:
[tex]\[
x = -4 \pm \sqrt{26}
\][/tex]
Thus, the solutions to the equation [tex]\(x^2 + 8x = 10\)[/tex] are:
[tex]\[
x = -4 \pm \sqrt{26}
\][/tex]
Therefore, the correct answer is:
D. [tex]\(x = -4 \pm \sqrt{26}\)[/tex]
