Certainly! Let's solve the equation [tex]\( 5x = 10 - \frac{1}{x} \)[/tex].
### Step-by-Step Solution
1. Rewrite the equation:
The given equation is:
[tex]\[
5x = 10 - \frac{1}{x}
\][/tex]
2. Eliminate the fraction:
To eliminate the fraction, multiply both sides of the equation by [tex]\( x \)[/tex]:
[tex]\[
x \cdot 5x = x \cdot \left(10 - \frac{1}{x}\right)
\][/tex]
Simplify:
[tex]\[
5x^2 = 10x - 1
\][/tex]
3. Form a standard quadratic equation:
Rearrange the equation to get all terms on one side:
[tex]\[
5x^2 - 10x + 1 = 0
\][/tex]
4. Solve the quadratic equation:
This is a standard quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = 5 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = 1 \)[/tex]. Use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 5 \cdot 1}}{2 \cdot 5}
\][/tex]
Simplify further:
[tex]\[
x = \frac{10 \pm \sqrt{100 - 20}}{10}
\][/tex]
[tex]\[
x = \frac{10 \pm \sqrt{80}}{10}
\][/tex]
[tex]\[
x = \frac{10 \pm 4\sqrt{5}}{10}
\][/tex]
Simplify by dividing the numerator by 10:
[tex]\[
x = 1 \pm \frac{2\sqrt{5}}{5}
\][/tex]
5. Obtain the numerical values:
Evaluate the two possible solutions:
[tex]\[
x_1 = 1 + \frac{2\sqrt{5}}{5} \approx 1.894427190999916
\][/tex]
[tex]\[
x_2 = 1 - \frac{2\sqrt{5}}{5} \approx 0.10557280900008412
\][/tex]
### Conclusion
The solutions to the equation [tex]\( 5x = 10 - \frac{1}{x} \)[/tex] are approximately:
[tex]\[
x_1 \approx 1.894427190999916
\][/tex]
[tex]\[
x_2 \approx 0.10557280900008412
\][/tex]
These are the values of [tex]\( x \)[/tex] that satisfy the given equation.
