If one root of ` 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. `49//4`
B. `4//49`
C. 4
D. none of these

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript.

### Step 1: Use the given root to find p
We start with the quadratic equation:
\[ x^2 + px + 12 = 0 \]
We know that one of the roots is 4. Therefore, substituting \( x = 4 \) into the equation gives us:
\[ 4^2 + 4p + 12 = 0 \]

Calculating \( 4^2 \) and \( 12 \):
\[ 16 + 4p + 12 = 0 \]
Combining the constants:
\[ 28 + 4p = 0 \]

Now, we can isolate \( p \):
\[ 4p = -28 \]
\[ p = -7 \]

### Step 2: Substitute p into the second equation
Now we have found that \( p = -7 \). We substitute \( p \) into the second equation:
\[ x^2 + px + q = 0 \]
This becomes:
\[ x^2 - 7x + q = 0 \]

### Step 3: Set the condition for equal roots
For the equation to have equal roots, the discriminant must be zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
In our case, \( a = 1 \), \( b = -7 \), and \( c = q \):
\[ D = (-7)^2 - 4(1)(q) = 0 \]

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

### Step 4: Solve for q
Now we can solve for \( q \):
\[ 49 = 4q \]
Dividing both sides by 4:
\[ q = \frac{49}{4} \]

### Final Answer
Thus, the value of \( q \) is:
\[ q = \frac{49}{4} \]

---