Given the equation:
[tex]\[
\frac{x}{x-1} + \frac{4}{x} = 4,
\][/tex]
we need to find the solutions for \( x \).
Let's start solving the equation step by step:
### Step 1: Eliminate the fractions.
The equation contains fractions, so the first step is to eliminate them by finding a common denominator. The denominators here are \( x \) and \( x - 1 \), so the common denominator is \( x(x - 1) \).
Multiply every term by this common denominator:
[tex]\[
x(x - 1) \left( \frac{x}{x-1} \right) + x(x-1) \left( \frac{4}{x} \right) = 4 x(x-1)
\][/tex]
### Step 2: Simplify the equation.
Simplify each term:
[tex]\[
x^2 + 4(x-1) = 4x(x - 1)
\][/tex]
Distribute the terms:
[tex]\[
x^2 + 4x - 4 = 4x^2 - 4x
\][/tex]
### Step 3: Move all terms to one side of the equation.
Bring all terms to one side such that the equation equals zero:
[tex]\[
x^2 + 4x - 4 - 4x^2 + 4x = 0
\][/tex]
Combine like terms:
[tex]\[
-x^2 + 8x - 4 = 0
\][/tex]
This can be written as:
[tex]\[
-x^2 + 8x - 4 = 0
\][/tex]
or multiplying by -1 to simplify:
[tex]\[
x^2 - 8x + 4 = 0
\][/tex]
### Step 4: Solve the quadratic equation.
This is a standard quadratic equation of the form \( ax^2 + bx + c = 0 \). We can use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, \( a = 1 \), \( b = -8 \), and \( c = 4 \). Substitute these values into the quadratic formula:
[tex]\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(4)}}{2(1)}
\][/tex]
Simplify inside the square root:
[tex]\[
x = \frac{8 \pm \sqrt{64 - 16}}{2}
\][/tex]
[tex]\[
x = \frac{8 \pm \sqrt{48}}{2}
\][/tex]
Simplify the square root:
[tex]\[
x = \frac{8 \pm 4\sqrt{3}}{2}
\][/tex]
[tex]\[
x = 4 \pm 2\sqrt{3}
\][/tex]
### Step 5: Convert approximate values.
We know that \( \sqrt{3} \approx 1.732 \):
So,
[tex]\[
x \approx 4 \pm 2 \cdot 1.732
\][/tex]
[tex]\[
x \approx 4 \pm 3.464
\][/tex]
This gives us two solutions:
[tex]\[
x \approx 4 + 3.464 \approx 7.464
\][/tex]
[tex]\[
x \approx 4 - 3.464 \approx 0.536
\][/tex]
For the given choices:
### Comparison:
The numerical solutions closest to the calculated values (converted to exact forms) would be:
[tex]\[
x \approx \frac{2}{3} \approx 0.667
\][/tex]
and
[tex]\[
x = 2
\][/tex]
Thus, the correct answer is:
b) [tex]\( \frac{2}{3}, 2 \)[/tex]
