Expand the quotient by partial fractions. (x^2 -2x +30)/(x^2 - 10x +24)

Sure, here is the step-by-step solution:

Step 1: Factorize the denominator The denominator x^2 - 10x + 24 can be factored into (x - 4)(x - 6).

Step 2: Set up the partial fractions We can now express the original expression as the sum of two simpler fractions: (x^2 -2x +30)/(x^2 - 10x +24) = A/(x - 4) + B/(x - 6)

Step 3: Clear the fractions Multiply through by the common denominator to get rid of the fractions: x^2 -2x +30 = A(x - 6) + B(x - 4)

Step 4: Solve for A and B To solve for A and B, we can choose convenient values for x. For example, if we let x = 4, the second term on the right side will become zero, and we can solve for A: 16 - 8 + 30 = A(4 - 6) 38 = -2A A = -19

Similarly, if we let x = 6, the first term on the right side will become zero, and we can solve for B: 36 - 12 + 30 = B(6 - 4) 54 = 2B B = 27

Step 5: Write the final answer Substitute A and B back into the partial fractions: (x^2 -2x +30)/(x^2 - 10x +24) = -19/(x - 4) + 27/(x - 6)