If -2 is a root of the equation `3x^(2)+7x+p=0,` find the value of k so that the roots of the equation `x^(2)+k(4x+k-1)+p=0` are equal.

Options

A. `-2, 3/2`
B. `-1, 2/3`
C. `-5, 4/5`
D. `-3, 7/5`

Explanation

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

### Step 1: Determine the value of p
Given that -2 is a root of the equation \(3x^2 + 7x + p = 0\), we can substitute \(x = -2\) into the equation:

\[ 3(-2)^2 + 7(-2) + p = 0 \]

Calculating this gives:

\[ 3(4) - 14 + p = 0 \]

\[ 12 - 14 + p = 0 \]

\[ -2 + p = 0 \]

Thus, we find:

\[ p = 2 \]

### Step 2: Substitute p into the new equation
Now, we substitute \(p = 2\) into the second equation \(x^2 + k(4x + k - 1) + p = 0\):

\[ x^2 + k(4x + k - 1) + 2 = 0 \]

Expanding this gives:

\[ x^2 + 4kx + k^2 - k + 2 = 0 \]

### Step 3: Identify coefficients
In the standard form \(ax^2 + bx + c = 0\), we identify:

- \(a = 1\)
- \(b = 4k\)
- \(c = k^2 - k + 2\)

### Step 4: Condition for equal roots
For the roots of the quadratic equation to be equal, the discriminant must be zero:

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

Substituting the values of \(a\), \(b\), and \(c\):

\[ (4k)^2 - 4(1)(k^2 - k + 2) = 0 \]

Calculating this gives:

\[ 16k^2 - 4(k^2 - k + 2) = 0 \]

Expanding the second term:

\[ 16k^2 - 4k^2 + 4k - 8 = 0 \]

Combining like terms:

\[ 12k^2 + 4k - 8 = 0 \]

### Step 5: Simplify the equation
Dividing the entire equation by 4 to simplify:

\[ 3k^2 + k - 2 = 0 \]

### Step 6: Factor the quadratic equation
Now we will factor \(3k^2 + k - 2\):

To factor, we look for two numbers that multiply to \(3 \times -2 = -6\) and add to \(1\). The numbers \(3\) and \(-2\) work:

\[ 3k^2 + 3k - 2k - 2 = 0 \]

Grouping:

\[ 3k(k + 1) - 2(k + 1) = 0 \]

Factoring out \((k + 1)\):

\[ (k + 1)(3k - 2) = 0 \]

### Step 7: Solve for k
Setting each factor to zero gives:

1. \(k + 1 = 0 \Rightarrow k = -1\)
2. \(3k - 2 = 0 \Rightarrow k = \frac{2}{3}\)

### Final Answer
The values of \(k\) such that the roots of the equation are equal are:

\[ k = -1 \quad \text{or} \quad k = \frac{2}{3} \]