Let's find the solutions to the equation [tex]\( x^2 - x = 10 \)[/tex] by completing the square. Here’s a detailed step-by-step solution:
1. Rewrite the equation in the standard form for completing the square:
[tex]\[
x^2 - x - 10 = 0
\][/tex]
2. Isolate the constant term to one side:
[tex]\[
x^2 - x = 10
\][/tex]
3. Add and subtract the square of half the coefficient of [tex]\( x \)[/tex] inside the left side to complete the square. The coefficient of [tex]\( x \)[/tex] is [tex]\(-1\)[/tex].
[tex]\[
\left( \frac{-1}{2} \right)^2 = \left( \frac{-1}{2} \right)^2 = \frac{(-1)^2}{2^2} = \frac{1}{4}
\][/tex]
4. Add and subtract [tex]\(\frac{1}{4}\)[/tex] inside the equation to maintain equality:
[tex]\[
x^2 - x + \frac{1}{4} - \frac{1}{4} = 10
\][/tex]
Simplify the left side:
[tex]\[
\left( x - \frac{1}{2} \right)^2 - \frac{1}{4} = 10
\][/tex]
5. Move the subtracted [tex]\(\frac{1}{4}\)[/tex] to the other side of the equation:
[tex]\[
\left( x - \frac{1}{2} \right)^2 = 10 + \frac{1}{4}
\][/tex]
6. Combine the terms on the right side:
[tex]\[
10 + \frac{1}{4} = 10.25
\][/tex]
7. The equation now is:
[tex]\[
\left( x - \frac{1}{2} \right)^2 = 10.25
\][/tex]
8. Take the square root of both sides, remembering to consider both the positive and negative square roots:
[tex]\[
x - \frac{1}{2} = \pm \sqrt{10.25}
\][/tex]
Simplify the square root:
[tex]\[
\sqrt{10.25} \approx 3.2015621187164243
\][/tex]
9. Write the two possible solutions:
[tex]\[
x - \frac{1}{2} = 3.2015621187164243 \quad \text{or} \quad x - \frac{1}{2} = -3.2015621187164243
\][/tex]
10. Solve for [tex]\( x \)[/tex] in each case:
[tex]\[
x = \frac{1}{2} + 3.2015621187164243 = 2.7015621187164243
\][/tex]
[tex]\[
x = \frac{1}{2} - 3.2015621187164243 = -3.7015621187164243
\][/tex]
Thus, the solutions to the equation [tex]\( x^2 - x = 10 \)[/tex] are:
[tex]\[
x = 2.7015621187164243 \quad \text{and} \quad x = -3.7015621187164243
\][/tex]
