Answer:
Below
Explanation:
To determine the constants A, B, and C in the equation:
x + 36/x(x − 6)² = A/x + B/x − 6 + C/(x − 6)²
We can use partial fraction decomposition to break down the left side of the equation into three fractions with denominators x, x - 6, and (x - 6)². The form of the decomposition will be:
x + 36/x(x − 6)² = A/x + B/(x - 6) + C/(x - 6)²
Now, let's find A, B, and C:
First, clear the denominators by multiplying both sides of the equation by the common denominator, which is x(x - 6)²:
x + 36 = A(x - 6)² + Bx(x - 6) + C
Now, we can solve for A, B, and C by selecting appropriate values of x that will eliminate some of the terms:
1. Setting x = 6 will eliminate the B term:
6 + 36 = A(6 - 6)² + C
42 = A
2. Setting x = 0 will eliminate the C term:
0 + 36 = A(0 - 6)² + B(0)(0 - 6) + C
36 = 36C
Now, solve for C:
C = 36/36
C = 1
3. Setting x = any value other than 0 or 6 will eliminate the A and C terms:
For example, let's set x = 1:
1 + 36 = A(1 - 6)² + B(1)(1 - 6) + 1
37 = A(25) - 5B
Now, use the value of A (which we found to be 42):
37 = 42(25) - 5B
37 = 1050 - 5B
Now, solve for B:
5B = 1050 - 37
5B = 1013
B = 1013/5
B = 202.6 (approximately)
So, the constants A, B, and C are:
A = 42
B = 202.6 (approximately)
C = 1