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If (x^(2)+5)/(x^(2)+2)^(2)=1/(x^(2)+2)+k...

If `(x^(2)+5)/(x^(2)+2)^(2)=1/(x^(2)+2)+k/(x^(2)+2)^(2) " then " k=`

A

1

B

2

C

3

D

4

Text Solution

AI Generated Solution

The correct Answer is:
To solve the equation \[ \frac{x^2 + 5}{(x^2 + 2)^2} = \frac{1}{x^2 + 2} + \frac{k}{(x^2 + 2)^2}, \] we will follow these steps: ### Step 1: Write the Right Side with a Common Denominator The right-hand side of the equation has two fractions. To combine them, we need a common denominator, which is \((x^2 + 2)^2\). \[ \frac{1}{x^2 + 2} = \frac{(x^2 + 2)}{(x^2 + 2)^2} \] Now, we can rewrite the right side: \[ \frac{(x^2 + 2) + k}{(x^2 + 2)^2} \] ### Step 2: Set the Numerators Equal Since the denominators are equal, we can set the numerators equal to each other: \[ x^2 + 5 = (x^2 + 2) + k \] ### Step 3: Simplify the Equation Now, simplify the equation: \[ x^2 + 5 = x^2 + 2 + k \] Subtract \(x^2\) from both sides: \[ 5 = 2 + k \] ### Step 4: Solve for \(k\) Now, isolate \(k\): \[ k = 5 - 2 \] \[ k = 3 \] ### Final Answer Thus, the value of \(k\) is \[ \boxed{3} \]
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