Find the value of p for which the quadratic equation `(p+1)x^2+ -6(p+1)x+3 (p +9)=0, p != -1` has equal roots. Hence, find the roots of the equation.

Answer

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

\[ (p+1)x^2 - 6(p+1)x + 3(p + 9) = 0 \]

has equal roots, we need to use the condition that the discriminant \( D \) of the quadratic equation must be equal to zero. The discriminant \( D \) is given by the formula:

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

where \( a = p + 1 \), \( b = -6(p + 1) \), and \( c = 3(p + 9) \).

### Step 1: Calculate the Discriminant
Substituting the values of \( a \), \( b \), and \( c \) into the discriminant formula:

\[ D = [-6(p + 1)]^2 - 4(p + 1)[3(p + 9)] \]

Calculating \( D \):

\[ D = 36(p + 1)^2 - 12(p + 1)(p + 9) \]

### Step 2: Expand the Terms
Now, let's expand both terms in the discriminant:

1. For \( 36(p + 1)^2 \):
\[ 36(p^2 + 2p + 1) = 36p^2 + 72p + 36 \]

2. For \( 12(p + 1)(p + 9) \):
\[ 12(p^2 + 10p + 9) = 12p^2 + 120p + 108 \]

### Step 3: Combine the Terms
Now, substituting back into the discriminant:

\[ D = (36p^2 + 72p + 36) - (12p^2 + 120p + 108) \]

Combine like terms:

\[ D = (36p^2 - 12p^2) + (72p - 120p) + (36 - 108) \]
\[ D = 24p^2 - 48p - 72 \]

### Step 4: Set the Discriminant to Zero
For the roots to be equal, we set \( D = 0 \):

\[ 24p^2 - 48p - 72 = 0 \]

### Step 5: Simplify the Equation
Dividing the entire equation by 24:

\[ p^2 - 2p - 3 = 0 \]

### Step 6: Factor the Quadratic
Now, we can factor the quadratic:

\[ (p - 3)(p + 1) = 0 \]

### Step 7: Solve for \( p \)
Setting each factor to zero gives:

1. \( p - 3 = 0 \) → \( p = 3 \)
2. \( p + 1 = 0 \) → \( p = -1 \) (not valid as per the problem statement)

Thus, the only valid solution is:

\[ p = 3 \]

### Step 8: Find the Roots of the Quadratic Equation
Now, substituting \( p = 3 \) back into the original equation:

\[ (3 + 1)x^2 - 6(3 + 1)x + 3(3 + 9) = 0 \]
\[ 4x^2 - 24x + 36 = 0 \]

### Step 9: Simplify the Equation
Dividing the entire equation by 4:

\[ x^2 - 6x + 9 = 0 \]

### Step 10: Factor the Quadratic
Factoring gives:

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

### Step 11: Find the Roots
Thus, the root is:

\[ x = 3 \]

### Summary of the Solution
The value of \( p \) for which the quadratic equation has equal roots is \( p = 3 \), and the equal roots of the equation are \( x = 3 \).

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