 solve rational equation (x-4)/(-2)-(4x+5)/(2x+3)=0

To solve the rational equation:

\[
\frac{x - 4}{-2} - \frac{4x + 5}{2x + 3} = 0
\]

### Step 1: Simplify the first fraction
Rewrite \(\frac{x - 4}{-2}\) as:

\[
\frac{x - 4}{-2} = -\frac{x - 4}{2} = -\frac{x}{2} + 2
\]

Thus, the equation becomes:

\[
-\frac{x}{2} + 2 - \frac{4x + 5}{2x + 3} = 0
\]

### Step 2: Isolate the second fraction
Move the second fraction to the other side of the equation:

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

### Step 3: Clear the fractions
Multiply through by 2 to eliminate the denominator in \(-\frac{x}{2}\):

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

### Step 4: Simplify the equation
Now multiply the right-hand side:

\[
-x + 4 = \frac{8x + 10}{2x + 3}
\]

### Step 5: Multiply both sides by \(2x + 3\) to clear the denominator:

\[
(x - 4)(2x + 3) = 8x + 10
\]

### Step 6: Expand both sides
Expanding the left-hand side:

\[
(x - 4)(2x + 3) = 2x^2 + 3x - 8x - 12 = 2x^2 - 5x - 12
\]

So the equation becomes:

\[
2x^2 - 5x - 12 = 8x + 10
\]

### Step 7: Move all terms to one side
\[
2x^2 - 5x - 12 - 8x - 10 = 0
\]
\[
2x^2 - 13x - 22 = 0
\]

### Step 8: Solve the quadratic equation
We can solve \(2x^2 - 13x - 22 = 0\) using the quadratic formula:

\[
x = \frac{-(-13) \pm \sqrt{(-13)^2 - 4(2)(-22)}}{2(2)}
\]
\[
x = \frac{13 \pm \sqrt{169 + 176}}{4}
\]
\[
x = \frac{13 \pm \sqrt{345}}{4}
\]

Thus, the solutions are:

\[
x = \frac{13 \pm \sqrt{345}}{4}
\]

These are the exact solutions. You can leave the answer in this form or approximate the square root of 345 if needed.

Would you like any details or further explanation?

### Related questions:
1. How do you approximate solutions involving square roots?
2. Can quadratic equations have complex solutions?
3. What is the discriminant and how does it determine the nature of solutions?
4. How do you handle rational expressions when there are multiple fractions?
5. What are the common pitfalls when solving rational equations?

### Tip:
Always check for extraneous solutions when solving rational equations by substituting the solutions back into the original equation.

solve rational equation (x-4)/(-2)-(4x+5)/(2x+3)=0

This page provides a detailed solution to solving the rational equation (x-4)/(-2) - (4x+5)/(2x+3) = 0. Learn the step-by-step approach to simplifying rational expressions, isolating terms, and applying the quadratic formula. Perfect for students in Grades 9-12 looking to deepen their understanding of algebra and rational equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation