If the product of the root 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

Answer

To find the value of \( k \) in the equation \( x^2 - 5x + k = 15 \) given that the product of the roots is -3, we can follow these steps:

### Step 1: Rearranging the Equation
First, we need to rearrange the equation into standard quadratic form. We start with:
\[ x^2 - 5x + k = 15 \]
Subtract 15 from both sides to set the equation to zero:
\[ x^2 - 5x + (k - 15) = 0 \]

### Step 2: Identifying Coefficients
In the standard form of a quadratic equation \( ax^2 + bx + c = 0 \), we can identify:
- \( a = 1 \)
- \( b = -5 \)
- \( c = k - 15 \)

### Step 3: Using the Product of Roots Formula
The product of the roots \( \alpha \) and \( \beta \) of a quadratic equation is given by:
\[ \alpha \beta = \frac{c}{a} \]
Substituting the values we identified:
\[ \alpha \beta = \frac{k - 15}{1} = k - 15 \]

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

### Step 5: Solving for \( k \)
Now, we solve for \( k \):
\[ k - 15 = -3 \]
Adding 15 to both sides:
\[ k = -3 + 15 \]
\[ k = 12 \]

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

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