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
You're on a roll — Step 1: Factorize the denominator. The denominator is 25x - 24 - 6x^2. Rearrange it into standard quadratic form: -6x^2 + 25x - 24 Factor out -1: -(6x^2 - 25x + 24) Now, factorize the quadratic expression 6x^2 - 25x + 24. We look for two numbers that multiply to 6 × 24 = 144 and add up to -25. These numbers are -9 and -16. 6x^2 - 9x - 16x + 24 Factor by grouping: 3x(2x - 3) - 8(2x - 3) (3x - 8)(2x - 3) So, the denominator is -(3x - 8)(2x - 3), which can also be written as (8 - 3x)(2x - 3). The expression becomes: (2x-7)/((8-3x)(2x-3)) Step 2: Set up the partial fraction decomposition. We express the fraction as a sum of two simpler fractions: (2x-7)/((8-3x)(2x-3)) = (A)/(8-3x) + (B)/(2x-3) Multiply both sides by (8-3x)(2x-3) to clear the denominators: 2x-7 = A(2x-3) + B(8-3x) Step 3: Solve for the constants A and B. To find A, set 2x-3 = 0, which means x = (3)/(2): 2((3)/(2)) - 7 = A(2((3)/(2)) - 3) + B(8 - 3((3)/(2))) 3 - 7 = A(3 - 3) + B(8 - (9)/(2)) -4 = A(0) + B((16)/(2) - (9)/(2)) -4 = B((7)/(2)) B = -4 × (2)/(7) = -(8)/(7) To find B, set 8-3x = 0, which means 3x = 8, so x = (8)/(3): 2((8)/(3)) - 7 = A(2((8)/(3)) - 3) + B(8 - 3((8)/(3))) (16)/(3) - 7 = A((16)/(3) - (9)/(3)) + B(8 - 8) (16)/(3) - (21)/(3) = A((7)/(3)) + B(0) -(5)/(3) = A((7)/(3)) A = -(5)/(3) × (3)/(7) = -(5)/(7) Step 4: Write the final partial fraction expression. Substitute the values of A and B back into the partial fraction form: (2x-7)/(25x-24-6x^2) = (-5)/(7)8-3x + (-8)/(7)2x-3 This can be written as: -(5)/(7(8-3x)) - (8)/(7(2x-3)) Drop the next question.