This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Combine the fractions on the left side of the equation by finding a common denominator. The common denominator for x+1 and x is x(x+1). (2)/(x+1) - (3)/(x) = (3)/(2) (2 · x)/(x(x+1)) - (3 · (x+1))/(x(x+1)) = (3)/(2) (2x - 3(x+1))/(x(x+1)) = (3)/(2) Step 2: Distribute the -3 in the numerator and simplify the left side. (2x - 3x - 3)/(x(x+1)) = (3)/(2) (-x - 3)/(x(x+1)) = (3)/(2) Step 3: Cross-multiply to eliminate the denominators. 2(-x - 3) = 3x(x+1) Step 4: Expand both sides of the equation. -2x - 6 = 3x^2 + 3x Step 5: Rearrange the equation into the standard quadratic form ax^2 + bx + c = 0. 0 = 3x^2 + 3x + 2x + 6 3x^2 + 5x + 6 = 0 Step 6: Solve the quadratic equation using the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a. Here, a=3, b=5, and c=6. First, calculate the discriminant = b^2 - 4ac. = (5)^2 - 4(3)(6) = 25 - 72 = -47 Since the discriminant is negative, there are no real solutions. The solutions are complex numbers. Step 7: Find the complex solutions. x = -5 ± sqrt(-47)2(3) x = -5 ± isqrt(47)6 The solutions are: x_1 = -5 + isqrt(47)6 x_2 = -5 - isqrt(47)6 The solutions are x = -5 ± isqrt(47)6.