To solve the equation \( 5 + \frac{5}{x} - \frac{3}{2x} = \frac{3}{4x^2} \), let's go through the steps:

### Step 1: Multiply through by the least common denominator (LCD).
The LCD of the fractions on both sides is \( 4x^2 \). Multiply every term by \( 4x^2 \):

\[
4x^2 \times 5 + 4x^2 \times \frac{5}{x} - 4x^2 \times \frac{3}{2x} = 4x^2 \times \frac{3}{4x^2}
\]

### Step 2: Simplify each term.
\[
20x^2 + 20x - 6x = 3
\]

This simplifies to:
\[
20x^2 + 14x - 3 = 0
\]

### Step 3: Solve the quadratic equation.
Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 20 \), \( b = 14 \), and \( c = -3 \).

\[
x = \frac{-14 \pm \sqrt{14^2 - 4 \times 20 \times (-3)}}{2 \times 20}
\]

\[
x = \frac{-14 \pm \sqrt{196 + 240}}{40}
\]

\[
x = \frac{-14 \pm \sqrt{436}}{40}
\]

\[
x = \frac{-14 \pm 20.88}{40}
\]

So, the two possible solutions for \( x \) are:

\[
x_1 = \frac{6.88}{40} = 0.172
\]
\[
x_2 = \frac{-34.88}{40} = -0.872
\]

### Final Answer:
The solutions are approximately \( x = 0.172 \) and \( x = -0.872 \).

Would you like more details or have any questions?

### Related Questions:
1. How does the quadratic formula work, and when is it used?
2. Can you show how to simplify complex fractions before solving?
3. What are some methods for solving quadratic equations besides the quadratic formula?
4. How can we check if the solutions of a quadratic equation are correct?
5. What is the discriminant, and what does it tell us about the roots of a quadratic equation?

### Tip:
Always double-check your solutions by substituting them back into the original equation.

Learn how to solve the equation 5 + 5/x - 3/(2x) = 3/(4x^2) with a detailed step-by-step solution. This problem covers key concepts in equations and fractions, including the use of the quadratic formula for solving quadratic equations. Suitable for advanced high school students.

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