If the expression `(9x^(2))/(3x+5)` is written in the equivalent form `(25)/(3x+5)+k`, what is the k in terms of x?

Options

A. `9x^(2)`
B. `9x^(2)+5`
C. `3x-5`
D. `3x+5`

Answer

To solve the problem, we need to express \( k \) in terms of \( x \) given the equation:

\[ \frac{9x^2}{3x + 5} = \frac{25}{3x + 5} + k \]

### Step 1: Rearranging the Equation
We start by isolating \( k \):

\[ k = \frac{9x^2}{3x + 5} - \frac{25}{3x + 5} \]

### Step 2: Combining the Fractions
Since both terms on the right-hand side have the same denominator, we can combine them:

\[ k = \frac{9x^2 - 25}{3x + 5} \]

### Step 3: Factoring the Numerator
Next, we recognize that \( 9x^2 - 25 \) is a difference of squares, which can be factored:

\[ 9x^2 - 25 = (3x)^2 - (5)^2 = (3x + 5)(3x - 5) \]

### Step 4: Substituting Back into the Equation
Substituting this back into our expression for \( k \):

\[ k = \frac{(3x + 5)(3x - 5)}{3x + 5} \]

### Step 5: Canceling Common Factors
We can cancel \( 3x + 5 \) from the numerator and denominator (as long as \( 3x + 5 \neq 0 \)):

\[ k = 3x - 5 \]

### Final Answer
Thus, the value of \( k \) in terms of \( x \) is:

\[ k = 3x - 5 \]