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
Here's how to express the given fraction in partial fractions: The expression is (x^2+2)/(x^2-4). Step 1: Perform polynomial division or algebraic manipulation to make the fraction proper. Since the degree of the numerator is equal to the degree of the denominator, we first divide the numerator by the denominator. (x^2+2)/(x^2-4) = ((x^2-4)+6)/(x^2-4) = (x^2-4)/(x^2-4) + (6)/(x^2-4) = 1 + (6)/(x^2-4) Step 2: Factor the denominator of the proper fraction. The denominator x^2-4 is a difference of squares: x^2-4 = (x-2)(x+2) Step 3: Set up the partial fraction decomposition for the proper fraction (6)/(x^2-4). (6)/((x-2)(x+2)) = (A)/(x-2) + (B)/(x+2) Step 4: Solve for the constants A and B. Multiply both sides by (x-2)(x+2): 6 = A(x+2) + B(x-2) To find A, set x=2: 6 = A(2+2) + B(2-2) 6 = 4A + 0 4A = 6 A = (6)/(4) = (3)/(2) To find B, set x=-2: 6 = A(-2+2) + B(-2-2) 6 = 0 + B(-4) -4B = 6 B = (6)/(-4) = -(3)/(2) Step 5: Substitute the values of A and B back into the partial fraction form. (6)/(x^2-4) = (3)/(2)x-2 + (-3)/(2)x+2 = (3)/(2(x-2)) - (3)/(2(x+2)) Step 6: Combine with the integer part from Step 1. (x^2+2)/(x^2-4) = 1 + (3)/(2(x-2)) - (3)/(2(x+2)) The expression in partial fractions is 1 + (3)/(2(x-2)) - (3)/(2(x+2)). Last free one today — make it count tomorrow, or type /upgrade for unlimited.