Question

If the roots of the quadratic equation \( 2x^2 - 5x + k = 0 \) are real and distinct, what is the range of values for \( k \)?

• A. \( k>\frac{25}{8} \)
• B. \( k<\frac{25}{8} \)
• C. \( k>0 \)
• D. \( k<0 \)

Hint:

For real and distinct roots, the discriminant must be positive. If the discriminant is zero, the roots are real and equal. If the discriminant is negative, the roots are imaginary.

Answer 1

The Correct Option is B

For a quadratic equation to possess real and distinct roots, its discriminant must be a positive value. The discriminant, denoted by \( \Delta \), for a quadratic equation in the form \( ax^2 + bx + c = 0 \) is calculated as: \[ \Delta = b^2 - 4ac \] Considering the equation \( 2x^2 - 5x + k = 0 \), the coefficients are identified as: - \( a = 2 \), - \( b = -5 \), - \( c = k \). Substituting these into the discriminant formula yields: \[ \Delta = (-5)^2 - 4(2)(k) = 25 - 8k \] The condition for real and distinct roots necessitates that \( \Delta>0 \): \[ 25 - 8k>0 \] Rearranging this inequality gives: \[ 25>8k \] \[ k<\frac{25}{8} \] Therefore, the parameter \( k \) must be less than \( \frac{25}{8} \) for the equation to have real and distinct roots.