Sure, let’s solve the equation [tex]\( x^2 - 2x = 9 \)[/tex] by completing the square.
1. Start with the given equation:
[tex]\[
x^2 - 2x = 9
\][/tex]
2. Move the constant term to the other side to set up a standard quadratic form:
[tex]\[
x^2 - 2x - 9 = 0
\][/tex]
3. To complete the square, we first need to consider the quadratic expression [tex]\( x^2 - 2x \)[/tex].
4. Take the coefficient of [tex]\( x \)[/tex], divide by 2, and square it:
The coefficient of [tex]\( x \)[/tex] is [tex]\(-2\)[/tex].
[tex]\[
\left(\frac{-2}{2}\right)^2 = 1
\][/tex]
5. Add and subtract this square term inside the equation to maintain equality:
[tex]\[
x^2 - 2x = 9
\][/tex]
[tex]\[
x^2 - 2x + 1 - 1 = 9
\][/tex]
6. Rewrite the left side as a perfect square and simplify the equation:
[tex]\[
(x - 1)^2 - 1 = 9
\][/tex]
[tex]\[
(x - 1)^2 - 10 = 0
\][/tex]
7. Move the constant term [tex]\(-10\)[/tex] to the other side of the equation:
[tex]\[
(x - 1)^2 = 10
\][/tex]
8. Take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x - 1 = \pm \sqrt{10}
\][/tex]
9. Solve for [tex]\( x \)[/tex] by isolating it:
[tex]\[
x - 1 = \sqrt{10} \quad \text{or} \quad x - 1 = -\sqrt{10}
\][/tex]
[tex]\[
x = 1 + \sqrt{10} \quad \text{or} \quad x = 1 - \sqrt{10}
\][/tex]
10. So, the solutions are:
[tex]\[
x = 1 + \sqrt{10} \approx 4.162
\][/tex]
[tex]\[
x = 1 - \sqrt{10} \approx -2.162
\][/tex]
These are the solutions to the equation [tex]\( x^2 - 2x = 9 \)[/tex].
