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: Use the logarithm property a + b = (ab) to combine the terms on the left side of the equation. (x+2) + (2x-1) = ((x+2)(2x-1)) So the equation becomes: ((x+2)(2x-1)) = 15 Step 2: Since A = B implies A = B, equate the arguments of the logarithms. (x+2)(2x-1) = 15 Step 3: Expand the left side and rearrange the equation into a standard quadratic form ax^2 + bx + c = 0. 2x^2 - x + 4x - 2 = 15 2x^2 + 3x - 2 = 15 Subtract 15 from both sides: 2x^2 + 3x - 17 = 0 Step 4: Solve the quadratic equation using the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a. Here, a=2, b=3, c=-17. x = -3 ± sqrt(3^2 - 4(2)(-17))2(2) x = -3 ± sqrt(9 + 136)4 x = -3 ± sqrt(145)4 This gives two potential solutions: x_1 = -3 + sqrt(145)4 and x_2 = -3 - sqrt(145)4 Step 5: Check for extraneous solutions. The arguments of the original logarithms must be positive, i.e., x+2 > 0 and 2x-1 > 0. This means x > -2 and x > (1)/(2). So, we must have x > (1)/(2). Let's approximate sqrt(145). Since 12^2 = 144, sqrt(145) is slightly greater than 12, approximately 12.04. For x_1: x_1 = -3 + sqrt(145)4 ≈ (-3 + 12.04)/(4) = (9.04)/(4) = 2.26 This value is greater than (1)/(2), so it is a valid solution. For x_2: x_2 = -3 - sqrt(145)4 ≈ (-3 - 12.04)/(4) = (-15.04)/(4) = -3.76 This value is not greater than (1)/(2) (it's less than -2 and less than 1/2), so it is an extraneous solution and must be rejected. The only valid solution is: x = -3 + sqrt(145)4 Drop the next question.