To solve the equation [tex]\( 5x = 10 - \frac{1}{x} \)[/tex] by completing the square, we can follow these steps:
1. Get everything on one side of the equation so that the equation equals zero:
[tex]\[
5x - 10 + \frac{1}{x} = 0
\][/tex]
2. Multiply through by [tex]\( x \)[/tex] to eliminate the fraction, which makes the equation:
[tex]\[
5x^2 - 10x + 1 = 0
\][/tex]
3. Identify the coefficients to prepare for completing the square:
The coefficients for [tex]\( ax^2 + bx + c = 0 \)[/tex] are:
[tex]\[
a = 5, \quad b = -10, \quad c = 1
\][/tex]
4. Divide the whole equation by the coefficient of [tex]\( x^2 \)[/tex] (which is 5):
[tex]\[
x^2 - 2x + \frac{1}{5} = 0
\][/tex]
5. Move the constant term to the other side of the equation:
[tex]\[
x^2 - 2x = -\frac{1}{5}
\][/tex]
6. Complete the square:
To complete the square, take the coefficient of [tex]\( x \)[/tex], divide it by 2, and square it.
[tex]\[
\left( \frac{-2}{2} \right)^2 = 1
\][/tex]
Add and subtract this square inside the equation:
[tex]\[
x^2 - 2x + 1 - 1 = -\frac{1}{5}
\][/tex]
This simplifies to:
[tex]\[
(x - 1)^2 - 1 = -\frac{1}{5}
\][/tex]
7. Add 1 to both sides:
[tex]\[
(x - 1)^2 = -\frac{1}{5} + 1
\][/tex]
Simplify the right side:
[tex]\[
(x - 1)^2 = \frac{5}{5} - \frac{1}{5} = \frac{4}{5}
\][/tex]
8. Take the square root of both sides:
[tex]\[
x - 1 = \pm \sqrt{\frac{4}{5}}
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 1 \pm \frac{2}{\sqrt{5}}
\][/tex]
9. Rationalize the denominator (Optional):
[tex]\[
\frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}
\][/tex]
Therefore, the solutions are:
[tex]\[
x = 1 \pm \frac{2\sqrt{5}}{5}
\][/tex]
So, the solutions to the equation [tex]\( 5x = 10 - \frac{1}{x} \)[/tex] are:
[tex]\[
x = 1 - \frac{2\sqrt{5}}{5} \quad \text{and} \quad x = 1 + \frac{2\sqrt{5}}{5}
\][/tex]
