To solve the equation [tex]\( x^2 - 8x = 3 \)[/tex] by completing the square, follow these steps:
1. Rewrite the equation in standard form:
[tex]\[
x^2 - 8x - 3 = 0
\][/tex]
2. Move the constant term to the right side of the equation:
[tex]\[
x^2 - 8x = 3
\][/tex]
3. Add and subtract the same value to complete the square on the left side. To complete the square for the term involving [tex]\( x \)[/tex], we take half of the coefficient of [tex]\( x \)[/tex], square it, and add it to both sides. For the term [tex]\( -8x \)[/tex], half of [tex]\(-8\)[/tex] is [tex]\(-4\)[/tex] and [tex]\((-4)^2 = 16\)[/tex]:
[tex]\[
x^2 - 8x + 16 - 16 = 3
\][/tex]
4. Add [tex]\( 16 \)[/tex] to both sides to balance the equation:
[tex]\[
x^2 - 8x + 16 = 3 + 16
\][/tex]
5. Simplify the right side:
[tex]\[
x^2 - 8x + 16 = 19
\][/tex]
6. Express the left side as a perfect square:
[tex]\[
(x - 4)^2 = 19
\][/tex]
7. Take the square root of both sides:
[tex]\[
x - 4 = \pm \sqrt{19}
\][/tex]
8. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 4 \pm \sqrt{19}
\][/tex]
So, the solutions to the equation [tex]\( x^2 - 8x = 3 \)[/tex] are:
[tex]\[
x = 4 - \sqrt{19} \quad \text{and} \quad x = 4 + \sqrt{19}
\][/tex]
Therefore, the solution set is:
[tex]\[
\{4 - \sqrt{19}, 4 + \sqrt{19}\}
\][/tex]
Thus, the correct solution set from the given options is:
[tex]\[
\{4-\sqrt{19}, 4+\sqrt{19}\}
\][/tex]
