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

Options

A. `9//4`
B. 1
C. 2.5
D. 3

Answer

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

### Step 1: Substitute the root into the first equation
We know that \(-4\) is a root of the equation \(x^2 - px - 4 = 0\). This means that if we substitute \(x = -4\) into the equation, it should satisfy the equation.

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

### Step 2: Simplify the equation
Now, we will simplify the equation:

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

This simplifies to:

\[ 12 + 4p = 0 \]

### Step 3: Solve for \(p\)
Now, we can isolate \(p\):

\[ 4p = -12 \]
\[ p = -3 \]

### Step 4: Use the value of \(p\) in the second equation
Now that we have found \(p = -3\), we will substitute this value into the second quadratic equation \(x^2 - px + k = 0\):

\[ x^2 - (-3)x + k = 0 \implies x^2 + 3x + k = 0 \]

### Step 5: Set up the condition for equal roots
For the quadratic equation to have equal roots, the discriminant must be equal to zero. The discriminant \(\Delta\) for the equation \(ax^2 + bx + c = 0\) is given by:

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

In our case, \(a = 1\), \(b = 3\), and \(c = k\). Thus, the discriminant becomes:

\[ \Delta = 3^2 - 4(1)(k) = 9 - 4k \]

### Step 6: Set the discriminant equal to zero
For the roots to be equal, we set the discriminant to zero:

\[ 9 - 4k = 0 \]

### Step 7: Solve for \(k\)
Now, we can solve for \(k\):

\[ 4k = 9 \]
\[ k = \frac{9}{4} \]

### Final Answer
The value of \(k\) is \(\frac{9}{4}\).

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