Find the solution for the rational equation: \( ((2 /(x+1)+5 / 2 x)=2 \backslash) \). Answer:

To solve the rational equation \(\frac{2}{x+1} + \frac{5}{2x} = 2\), first, we find a common denominator for the fractions. The common denominator is \(2x(x+1)\). We rewrite the equation: \[ \frac{2 \cdot 2x}{2x(x+1)} + \frac{5(x+1)}{2x(x+1)} = 2 \] This simplifies to: \[ \frac{4x + 5(x+1)}{2x(x+1)} = 2 \] Combine the terms in the numerator: \[ \frac{4x + 5x + 5}{2x(x+1)} = 2 \rightarrow \frac{9x + 5}{2x(x+1)} = 2 \] Now, multiply both sides by \(2x(x+1)\): \[ 9x + 5 = 4x(x+1) \] Expanding the right side gives: \[ 9x + 5 = 4x^2 + 4x \] Rearranging all terms to one side results in: \[ 4x^2 - 5x - 5 = 0 \] Next, we apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\): For \(a = 4\), \(b = -5\), \(c = -5\): \[ x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 4 \cdot (-5)}}{2 \cdot 4} \] Calculating the discriminant: \[ x = \frac{5 \pm \sqrt{25 + 80}}{8} = \frac{5 \pm \sqrt{105}}{8} \] Thus, the solutions are: \[ x = \frac{5 + \sqrt{105}}{8} \quad \text{and} \quad x = \frac{5 - \sqrt{105}}{8} \] These are the solutions for the rational equation.