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
Fresh day BILIJITIN, let's solve. Step 1: Determine the domain of the variables. For the square roots to be defined, the expressions under them must be non-negative: 3x+4 0 3x -4 x -(4)/(3) x+5 0 x -5 Combining these conditions, we must have x -(4)/(3). Step 2: Isolate one of the radical terms. The given equation is sqrt(3x+4) - sqrt(x+5) = 1. Move the second radical term to the right side of the equation: sqrt(3x+4) = 1 + sqrt(x+5) Step 3: Square both sides of the equation. (sqrt(3x+4))^2 = (1 + sqrt(x+5))^2 3x+4 = 1^2 + 2(1)sqrt(x+5) + (sqrt(x+5))^2 3x+4 = 1 + 2sqrt(x+5) + x+5 3x+4 = x+6 + 2sqrt(x+5) Step 4: Isolate the remaining radical term. Subtract x+6 from both sides: 3x+4 - x - 6 = 2sqrt(x+5) 2x - 2 = 2sqrt(x+5) Divide both sides by 2: x - 1 = sqrt(x+5) Step 5: Square both sides again. (x-1)^2 = (sqrt(x+5))^2 x^2 - 2x + 1 = x+5 Step 6: Rearrange the equation into a standard quadratic form ax^2+bx+c=0. x^2 - 2x - x + 1 - 5 = 0 x^2 - 3x - 4 = 0 Step 7: Solve the quadratic equation for x. Factor the quadratic equation: (x-4)(x+1) = 0 This gives two possible solutions: x_1 = 4 x_2 = -1 Step 8: Check the solutions against the domain and the original equation. Recall the domain x -(4)/(3). Both x=4 and x=-1 satisfy this condition. Substitute each solution into the original equation sqrt(3x+4) - sqrt(x+5) = 1: For x=4: sqrt(3(4)+4) - sqrt(4+5) = sqrt(12+4) - sqrt(9) = sqrt(16) - 3 = 4 - 3 = 1 This matches the right side of the equation, so x=4 is a valid solution. For x=-1: sqrt(3(-1)+4) - sqrt(-1+5) = sqrt(-3+4) - sqrt(4) = sqrt(1) - 2 = 1 - 2 = -1 This does not match the right side of the equation (which is 1), so x=-1 is an extraneous solution. The only valid solution is x=4. The final answer is x=4. Drop the next question.