To solve the quadratic equation [tex]\( x^2 - 8x = 3 \)[/tex] by completing the square, follow these detailed steps:
1. Rewrite the equation in standard form:
[tex]\[
x^2 - 8x - 3 = 0
\][/tex]
2. Move the constant term to the other side of the equation:
[tex]\[
x^2 - 8x = 3
\][/tex]
3. Complete the square:
- Take half of the coefficient of [tex]\(x\)[/tex], which is [tex]\(-8\)[/tex]:
[tex]\[
\frac{-8}{2} = -4
\][/tex]
- Square it:
[tex]\[
(-4)^2 = 16
\][/tex]
- Add and subtract this square to the left side:
[tex]\[
x^2 - 8x + 16 - 16 = 3
\][/tex]
[tex]\[
(x - 4)^2 - 16 = 3
\][/tex]
4. Isolate the squared term by moving the constant to the other side:
[tex]\[
(x - 4)^2 - 16 = 3
\][/tex]
[tex]\[
(x - 4)^2 = 3 + 16
\][/tex]
[tex]\[
(x - 4)^2 = 19
\][/tex]
5. Take the square root of both sides:
[tex]\[
x - 4 = \pm \sqrt{19}
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = 4 \pm \sqrt{19}
\][/tex]
Therefore, the solutions to the equation [tex]\( x^2 - 8x = 3 \)[/tex] are [tex]\( 4 - \sqrt{19} \)[/tex] and [tex]\( 4 + \sqrt{19} \)[/tex].
Given the options:
- [tex]\( \{4 - \sqrt{19}, 4 + \sqrt{19}\} \)[/tex]
The solution set is:
[tex]\[
\boxed{\{4 - \sqrt{19}, 4 + \sqrt{19}\}}
\][/tex]
The numerical values, rounded appropriately, are approximately [tex]\(-0.3589 \)[/tex] and [tex]\( 8.3589 \)[/tex].
