If `-4` is a root of the equation `x^(2)+px-4=0` and the equation `x^(2)+px+q=0` has equal roots, then the value of q is

Options

A. `-(9)/(4)`
B. `(9)/(4)`
C. `(4)/(9)`
D. 36

Answer

To solve the problem, we will follow these steps:

1. **Identify the first equation**: We are given the equation \( x^2 + px - 4 = 0 \) and that \(-4\) is a root of this equation.

2. **Use the fact that -4 is a root**: If \(-4\) is a root, we can substitute \(x = -4\) into the equation:
\[ (-4)^2 + p(-4) - 4 = 0 \]
Simplifying this gives:
\[ 16 - 4p - 4 = 0 \implies 12 - 4p = 0 \implies 4p = 12 \implies p = 3 \]

3. **Identify the second equation**: The second equation is \( x^2 + px + q = 0 \) which has equal roots. For a quadratic equation to have equal roots, the discriminant must be zero.

4. **Calculate the discriminant**: The discriminant \(D\) for the equation \(x^2 + px + q = 0\) is given by:
\[ D = p^2 - 4q \]
Setting the discriminant equal to zero for equal roots:
\[ p^2 - 4q = 0 \]

5. **Substitute the value of p**: We already found \(p = 3\), so we substitute this value into the discriminant equation:
\[ 3^2 - 4q = 0 \implies 9 - 4q = 0 \implies 4q = 9 \implies q = \frac{9}{4} \]

6. **Final result**: Therefore, the value of \(q\) is:
\[ q = \frac{9}{4} \]