If the product of the roots of the equation `x^(2) - 5x+ k = 15 " is " -3`, then what is the value of k?

Options

A. 12
B. 15
C. 16
D. 18

Explanation

To solve the problem step by step, we start with the given quadratic equation and the information about the product of its roots.

### Step 1: Write the quadratic equation
The given equation is:
\[ x^2 - 5x + k = 15 \]

### Step 2: Rearrange the equation
We can rearrange the equation to standard form:
\[ x^2 - 5x + (k - 15) = 0 \]
Here, we identify the coefficients:
- \( a = 1 \)
- \( b = -5 \)
- \( c = k - 15 \)

### Step 3: Use the product of the roots formula
The product of the roots (let's denote them as \( \alpha \) and \( \beta \)) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ \alpha \beta = \frac{c}{a} \]
Substituting our values:
\[ \alpha \beta = \frac{k - 15}{1} = k - 15 \]

### Step 4: Set the product of the roots equal to -3
According to the problem, the product of the roots is given as -3. Therefore, we can set up the equation:
\[ k - 15 = -3 \]

### Step 5: Solve for k
Now, we solve for \( k \):
\[ k - 15 = -3 \\ k = -3 + 15 \\ k = 12 \]

### Conclusion
Thus, the value of \( k \) is:
\[ k = 12 \]