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"Determine the constants A, B, and C. x + 36/x(x − 6)² = A/x+ B/x − 6 + C/(x − 6)² A = B = C ="

User Istari
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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

User Maulik Hirani
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