The equation `3x^(2)-12x+(n-5)=0` has repeated roots. Find the value of of n.

Options

A. 15
B. 16
C. 17
D. 18

Explanation

To find the value of \( n \) for which the quadratic equation \( 3x^2 - 12x + (n - 5) = 0 \) has repeated roots, we need to use the condition for repeated roots. This condition states that the discriminant of the quadratic equation must be equal to zero.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is in the form \( ax^2 + bx + c = 0 \).
Here, \( a = 3 \), \( b = -12 \), and \( c = n - 5 \).

2. **Write the discriminant formula**:
The discriminant \( D \) of a quadratic equation is given by:
\[ D = b^2 - 4ac \]

3. **Set the discriminant to zero for repeated roots**:
For the roots to be repeated, we set the discriminant equal to zero:
\[ b^2 - 4ac = 0 \]

4. **Substitute the values of \( a \), \( b \), and \( c \)**:
Substitute \( b = -12 \), \( a = 3 \), and \( c = n - 5 \) into the discriminant formula:
\[ (-12)^2 - 4 \cdot 3 \cdot (n - 5) = 0 \]

5. **Calculate \( b^2 \)**:
Calculate \( (-12)^2 \):
\[ 144 - 4 \cdot 3 \cdot (n - 5) = 0 \]

6. **Simplify the equation**:
Now simplify the equation:
\[ 144 - 12(n - 5) = 0 \]
Distributing \( -12 \):
\[ 144 - 12n + 60 = 0 \]
Combine like terms:
\[ 204 - 12n = 0 \]

7. **Solve for \( n \)**:
Rearranging gives:
\[ 12n = 204 \]
Dividing both sides by 12:
\[ n = \frac{204}{12} = 17 \]

### Final Answer:
Thus, the value of \( n \) is \( 17 \).

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