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

To solve the equation \( \frac{2}{x+1} + \frac{5}{2x} = 2 \), let's first eliminate the fractions by finding a common denominator, which will be \( 2x(x+1) \). Multiplying through by the common denominator, we get: \[ 2x \cdot 2 + 5(x+1) = 2 \cdot 2x(x+1) \] This simplifies to: \[ 4x + 5x + 5 = 4x^2 + 8x \] Combining like terms gives: \[ 9x + 5 = 4x^2 + 8x \] Rearranging everything to one side results in: \[ 4x^2 - x - 5 = 0 \] Next, we can solve this quadratic using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 4, b = -1, c = -5 \). Calculating the discriminant: \[ b^2 - 4ac = (-1)^2 - 4(4)(-5) = 1 + 80 = 81 \] Now applying the quadratic formula: \[ x = \frac{-(-1) \pm \sqrt{81}}{2 \cdot 4} = \frac{1 \pm 9}{8} \] This gives us two possible solutions: 1. \( x = \frac{10}{8} = \frac{5}{4} \) 2. \( x = \frac{-8}{8} = -1 \) However, we must check for extraneous solutions by substituting back into the original equation and noting that \( x = -1 \) makes the denominator of \( \frac{2}{x+1} \) undefined. Thus, the final solution is \( x = \frac{5}{4} \).