Find the value of p for which the quadratic equation `x^(2)+p(4x+p-1)+2=0` has equal roots :

Options

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

Answer

To find the value of \( p \) for which the quadratic equation

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

has equal roots, we will follow these steps:

### Step 1: Rewrite the Equation
First, we can rewrite the quadratic equation in standard form. Distributing \( p \) gives us:

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

### Step 2: Identify Coefficients
From the standard form \( ax^2 + bx + c = 0 \), we identify the coefficients:
- \( a = 1 \)
- \( b = 4p \)
- \( c = p^2 - p + 2 \)

### Step 3: Set the Discriminant to Zero
For the quadratic equation to have equal roots, the discriminant must be equal to zero. The discriminant \( D \) is given by:

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

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

\[ D = (4p)^2 - 4(1)(p^2 - p + 2) \]

### Step 4: Simplify the Discriminant
Calculating \( D \):

\[ D = 16p^2 - 4(p^2 - p + 2) \]

Distributing the \( -4 \):

\[ D = 16p^2 - 4p^2 + 4 - 8 \]

Combining like terms:

\[ D = 12p^2 - 4 \]

### Step 5: Set the Discriminant to Zero
Setting the discriminant equal to zero for equal roots:

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

### Step 6: Solve for \( p \)
Rearranging the equation gives:

\[ 12p^2 = 4 \]

Dividing both sides by 12:

\[ p^2 = \frac{1}{3} \]

Taking the square root of both sides:

\[ p = \pm \frac{1}{\sqrt{3}} = \pm \frac{\sqrt{3}}{3} \]

### Step 7: Final Values of \( p \)
Thus, the values of \( p \) for which the quadratic equation has equal roots are:

\[ p = \frac{\sqrt{3}}{3} \quad \text{and} \quad p = -\frac{\sqrt{3}}{3} \]