If one root of the equation `x^(2) + px + 12 = 0` is 4, while the equation `x^(2)+ px + q = 0` has equal roots, then the value of 'q' is

Options

A. 3
B. 12
C. `(49)/(4)`
D. 16

Answer

To solve the problem step by step, we will follow the information given in the question.

### Step 1: Use the first equation to find 'p'
Given the equation:
\[ x^2 + px + 12 = 0 \]
We know that one root is \( x = 4 \). By substituting \( x = 4 \) into the equation, we can find the value of \( p \).

Substituting \( x = 4 \):
\[ 4^2 + p \cdot 4 + 12 = 0 \]
Calculating \( 4^2 \):
\[ 16 + 4p + 12 = 0 \]
Combining like terms:
\[ 4p + 28 = 0 \]
Now, isolate \( p \):
\[ 4p = -28 \]
\[ p = -7 \]

### Step 2: Use the second equation to find 'q'
Now we have \( p = -7 \). We will use this value in the second equation:
\[ x^2 + px + q = 0 \]
This equation has equal roots, which means the discriminant must be zero. The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
For our equation:
- \( a = 1 \)
- \( b = p = -7 \)
- \( c = q \)

Setting the discriminant to zero:
\[ D = (-7)^2 - 4 \cdot 1 \cdot q = 0 \]
Calculating \( (-7)^2 \):
\[ 49 - 4q = 0 \]
Now, isolate \( q \):
\[ 4q = 49 \]
\[ q = \frac{49}{4} \]

### Final Answer
The value of \( q \) is \( \frac{49}{4} \).

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