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. 4
B. `49//4`
C. `4//49`
D. None of these

Answer

To solve the problem step by step, we need to find the value of \( q \) given the conditions of the quadratic equations.

### Step 1: Identify the first equation and its root
The first equation is given as:

\[ x^2 + px + 12 = 0 \]

We know that one root, \( \alpha \), is 4. Let's denote the other root as \( \beta \).

### Step 2: Use the relationship between the roots and coefficients
From Vieta's formulas, we know:

1. The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
2. The product of the roots \( \alpha \cdot \beta = \frac{c}{a} \)

For our equation, \( a = 1 \), \( b = p \), and \( c = 12 \).

Thus, we have:

\[ \alpha + \beta = -p \quad (1) \]
\[ \alpha \cdot \beta = 12 \quad (2) \]

### Step 3: Substitute the known root into the equations
Substituting \( \alpha = 4 \) into equation (1):

\[ 4 + \beta = -p \quad \Rightarrow \quad \beta = -p - 4 \quad (3) \]

Substituting \( \alpha = 4 \) into equation (2):

\[ 4 \cdot \beta = 12 \quad \Rightarrow \quad \beta = \frac{12}{4} = 3 \quad (4) \]

### Step 4: Find the value of \( p \)
Now, we can substitute \( \beta = 3 \) from equation (4) into equation (3):

\[ 3 = -p - 4 \quad \Rightarrow \quad -p = 3 + 4 \quad \Rightarrow \quad -p = 7 \quad \Rightarrow \quad p = -7 \]

### Step 5: Analyze the second equation
The second equation is given as:

\[ x^2 + px + q = 0 \]

We need to find \( q \) such that this equation has equal roots. For a quadratic equation to have equal roots, the discriminant must be zero:

\[ D = b^2 - 4ac = 0 \]

Substituting \( a = 1 \), \( b = p \), and \( c = q \):

\[ p^2 - 4q = 0 \quad (5) \]

### Step 6: Substitute the value of \( p \) into the discriminant equation
Now, substituting \( p = -7 \) into equation (5):

\[ (-7)^2 - 4q = 0 \]
\[ 49 - 4q = 0 \]

### Step 7: Solve for \( q \)
Rearranging the equation gives:

\[ 4q = 49 \quad \Rightarrow \quad q = \frac{49}{4} \]

### Final Answer
Thus, the value of \( q \) is:

\[ \boxed{\frac{49}{4}} \]