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

Solution Steps

To find the value of \( p \) for which the quadratic equation \( p x^{2} - 4 p x + 2 - p = 0 \) has equal roots, we need to use the condition for equal roots in a quadratic equation. The condition is that the discriminant (\( \Delta \)) must be zero. The discriminant for a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \Delta = b^2 - 4ac \). We will set this discriminant to zero and solve for \( p \).

Given the quadratic equation: \[ p x^{2} - 4 p x + 2 - p = 0 \]

For a quadratic equation \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is given by: \[ \Delta = b^2 - 4ac \] Here, \( a = p \), \( b = -4p \), and \( c = 2 - p \). Substituting these values, we get: \[ \Delta = (-4p)^2 - 4(p)(2 - p) \] \[ \Delta = 16p^2 - 4p(2 - p) \] \[ \Delta = 16p^2 - 8p + 4p^2 \] \[ \Delta = 20p^2 - 8p \]

For the quadratic equation to have equal roots, the discriminant must be zero: \[ 20p^2 - 8p = 0 \]

Factor the equation: \[ 4p(5p - 2) = 0 \] Setting each factor to zero gives: \[ 4p = 0 \quad \text{or} \quad 5p - 2 = 0 \] \[ p = 0 \quad \text{or} \quad p = \frac{2}{5} \]

Final Answer