Let's solve the equation \(\frac{2x}{x + 1} + \frac{5}{2x} = 2\).

### Step 1: Find a common denominator
To eliminate the fractions, find the common denominator of the terms on the left side. The common denominator is \((x + 1)(2x)\).

Multiply each term by \((x + 1)(2x)\) to eliminate the fractions:

\[
(x + 1)(2x) \times \frac{2x}{x + 1} + (x + 1)(2x) \times \frac{5}{2x} = (x + 1)(2x) \times 2
\]

Simplifying:

\[
2x \cdot 2x + 5(x + 1) = 2(x + 1)(2x)
\]

This simplifies to:

\[
4x^2 + 5(x + 1) = 4x(2x + 2)
\]

### Step 2: Expand both sides
Expand the terms on both sides:

\[
4x^2 + 5x + 5 = 8x^2 + 8x
\]

### Step 3: Rearrange the equation
Move all terms to one side of the equation:

\[
4x^2 + 5x + 5 - 8x^2 - 8x = 0
\]

Simplifying:

\[
-4x^2 - 3x + 5 = 0
\]

### Step 4: Multiply by \(-1\) to simplify the equation
\[
4x^2 + 3x - 5 = 0
\]

### Step 5: Solve the quadratic equation
Use the quadratic formula to solve for \(x\):

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where \(a = 4\), \(b = 3\), and \(c = -5\):

\[
x = \frac{-3 \pm \sqrt{3^2 - 4(4)(-5)}}{2(4)}
\]

Simplify:

\[
x = \frac{-3 \pm \sqrt{9 + 80}}{8} = \frac{-3 \pm \sqrt{89}}{8}
\]

So the solutions are:

\[
x = \frac{-3 + \sqrt{89}}{8} \quad \text{or} \quad x = \frac{-3 - \sqrt{89}}{8}
\]

These are the two possible values of \(x\).

Would

Learn how to solve the equation (2x)/(x + 1) + 5/(2x) = 2 with a detailed step-by-step solution. This problem covers algebraic equations involving fractions and requires finding a common denominator. Suitable for students in Grades 10-12.

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