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. `1//4`

Explanation

To solve the problem step by step, we will follow the given information and apply the properties of quadratic equations.

### Step 1: Use the first equation to find p
We know that one root of the equation \( x^2 + px + 12 = 0 \) is 4. We can substitute \( x = 4 \) into the equation to find the value of \( p \).

\[ 4^2 + 4p + 12 = 0 \]

Calculating \( 4^2 \):

\[ 16 + 4p + 12 = 0 \]

Combine the constants:

\[ 4p + 28 = 0 \]

Now, isolate \( p \):

\[ 4p = -28 \]

\[ p = -7 \]

### Step 2: Substitute p into the second equation
Now that we have \( p = -7 \), we can substitute this value into the second equation \( x^2 + px + q = 0 \):

\[ x^2 - 7x + q = 0 \]

### Step 3: Use the condition for equal roots
For the quadratic equation to have equal roots, the discriminant must be zero. The discriminant \( D \) for the equation \( ax^2 + bx + c = 0 \) is given by:

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

In our case, \( a = 1 \), \( b = -7 \), and \( c = q \):

\[ D = (-7)^2 - 4(1)(q) \]

Calculating \( (-7)^2 \):

\[ D = 49 - 4q \]

Setting the discriminant equal to zero for equal roots:

\[ 49 - 4q = 0 \]

### Step 4: Solve for q
Now, we can solve for \( q \):

\[ 49 = 4q \]

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

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

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

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