Question

For what value of \( p \), the roots of the quadratic equation \( px^2 - 2x + 3 = 0 \) are real and equal?

• A. \( 1 \)
• B. \( \frac{1}{3} \)
• C. \( 3 \)
• D. \( \frac{1}{2} \)

Hint:

The roots of a quadratic equation are equal if the discriminant \( D = b^2 - 4ac \) is zero.

Answer 1

The Correct Option is B

Step 1: Use the condition for equal roots For a quadratic equation \( ax^2 + bx + c = 0 \), the roots are real and equal if: \[ D = b^2 - 4ac = 0 \] Here, \( a = p \), \( b = -2 \), and \( c = 3 \). Step 2: Solve for \( p \) \[ (-2)^2 - 4(p)(3) = 0 \] \[ 4 - 12p = 0 \] \[ 12p = 4 \] \[ p = \frac{1}{3} \] Thus, the correct answer is \( \frac{1}{3} \).