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If (3x+2)/((x+1)(2x^(2)+3))=(A)/(x+1)-(B...

If `(3x+2)/((x+1)(2x^(2)+3))=(A)/(x+1)-(Bx+C)/(2x^(2)+3)`, then `A+B-C=`

A

0

B

2

C

3

D

5

Text Solution

AI Generated Solution

The correct Answer is:
To solve the equation \[ \frac{3x + 2}{(x + 1)(2x^2 + 3)} = \frac{A}{x + 1} - \frac{Bx + C}{2x^2 + 3}, \] we will follow these steps: ### Step 1: Combine the Right Side We start by finding a common denominator for the right side: \[ \frac{A}{x + 1} - \frac{Bx + C}{2x^2 + 3} = \frac{A(2x^2 + 3) - (Bx + C)(x + 1)}{(x + 1)(2x^2 + 3)}. \] ### Step 2: Set Numerators Equal Since the denominators are the same, we can equate the numerators: \[ 3x + 2 = A(2x^2 + 3) - (Bx + C)(x + 1). \] ### Step 3: Expand the Right Side Now, we expand the right side: \[ 3x + 2 = A(2x^2 + 3) - (Bx^2 + Bx + Cx + C). \] This simplifies to: \[ 3x + 2 = 2Ax^2 + 3A - Bx^2 - (B + C)x - C. \] ### Step 4: Combine Like Terms Combine like terms on the right side: \[ 3x + 2 = (2A - B)x^2 + (3A - (B + C))x + (-C). \] ### Step 5: Compare Coefficients Now we compare the coefficients from both sides of the equation: 1. Coefficient of \(x^2\): \(0 = 2A - B\) (1) 2. Coefficient of \(x\): \(3 = 3A - (B + C)\) (2) 3. Constant term: \(2 = -C\) (3) ### Step 6: Solve the Equations From equation (3), we find: \[ C = -2. \] Substituting \(C = -2\) into equation (2): \[ 3 = 3A - (B - 2) \implies 3 = 3A - B + 2 \implies 1 = 3A - B \implies B = 3A - 1. \quad (4) \] Now, substitute equation (4) into equation (1): \[ 0 = 2A - (3A - 1) \implies 0 = 2A - 3A + 1 \implies A = 1. \] Now substitute \(A = 1\) back into equation (4): \[ B = 3(1) - 1 = 2. \] ### Step 7: Calculate \(A + B - C\) Now we have: - \(A = 1\) - \(B = 2\) - \(C = -2\) Now calculate \(A + B - C\): \[ A + B - C = 1 + 2 - (-2) = 1 + 2 + 2 = 5. \] ### Final Answer Thus, the value of \(A + B - C\) is \[ \boxed{5}. \]
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