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
This problem involves partial fraction decomposition, which is a method to break down a complex rational expression into simpler fractions. Step 1: Set up the partial fraction decomposition. The denominator has two distinct linear factors: (2x - 1) and (x + 2). Therefore, we can write the expression as a sum of two fractions with these factors as denominators and unknown constants (A and B) as numerators. (5x + 3) / ((2x - 1)(x + 2)) = A/(2x - 1) + B/(x + 2) Step 2: Combine the partial fractions. To solve for A and B, we first find a common denominator for the right side, which is (2x - 1)(x + 2). A/(2x - 1) + B/(x + 2) = (A(x + 2) + B(2x - 1)) / ((2x - 1)(x + 2)) Step 3: Equate the numerators. Since the denominators are now the same, the numerators must be equal. 5x + 3 = A(x + 2) + B(2x - 1) Step 4: Solve for A and B. We can find A and B by choosing specific values for x that simplify the equation. To find A, let 2x - 1 = 0. This means x = 1/2. Substitute x = 1/2 into the equation: 5(1/2) + 3 = A(1/2 + 2) + B(2(1/2) - 1) 5/2 + 6/2 = A(1/2 + 4/2) + B(1 - 1) 11/2 = A(5/2) + B(0) 11/2 = 5A/2 Multiply both sides by 2: 11 = 5A A = 11/5 To find B, let x + 2 = 0. This means x = -2. Substitute x = -2 into the equation: 5(-2) + 3 = A(-2 + 2) + B(2(-2) - 1) -10 + 3 = A(0) + B(-4 - 1) -7 = B(-5) -7 = -5B B = (-7)/(-5) B = 7/5 Step 5: Write the final partial fraction decomposition. Substitute the values of A and B back into the setup from Step 1. (5x + 3) / ((2x - 1)(x + 2)) = (11/5)/(2x - 1) + (7/5)/(x + 2) This can also be written as: = 11 / (5(2x - 1)) + 7 / (5(x + 2))