The two roots of equation `x^(2) - px + 8p - 15 = 0` are equal, then the value of `p` will be

Options

A. 3 or 5
B. 2 or 5
C. 3 or 4
D. 2 or 30

Answer

To solve the equation \( x^2 - px + 8p - 15 = 0 \) for the value of \( p \) when the roots are equal, we can follow these steps:

### Step 1: Understand the condition for equal roots
For a quadratic equation \( ax^2 + bx + c = 0 \), the roots are equal if the discriminant \( D \) is zero. The discriminant is given by:
\[ D = b^2 - 4ac \]
In our case, \( a = 1 \), \( b = -p \), and \( c = 8p - 15 \).

### Step 2: Set up the discriminant
Substituting the values of \( a \), \( b \), and \( c \) into the discriminant formula:
\[ D = (-p)^2 - 4(1)(8p - 15) \]
This simplifies to:
\[ D = p^2 - 4(8p - 15) \]

### Step 3: Simplify the discriminant
Now, simplify the expression:
\[ D = p^2 - 32p + 60 \]

### Step 4: Set the discriminant to zero
For the roots to be equal, we set the discriminant \( D \) to zero:
\[ p^2 - 32p + 60 = 0 \]

### Step 5: Solve the quadratic equation
We can solve this quadratic equation using the factorization method or the quadratic formula. Let's factor it:
We need two numbers that multiply to \( 60 \) and add up to \( -32 \). The numbers are \( -30 \) and \( -2 \):
\[ (p - 30)(p - 2) = 0 \]

### Step 6: Find the values of \( p \)
Setting each factor to zero gives us:
\[ p - 30 = 0 \quad \Rightarrow \quad p = 30 \]
\[ p - 2 = 0 \quad \Rightarrow \quad p = 2 \]

### Conclusion
The values of \( p \) for which the roots of the equation are equal are \( p = 2 \) and \( p = 30 \).