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 the roots of a quadratic equation to be real and distinct, the discriminant must be positive. The discriminant \( \Delta \) of the quadratic equation \( ax^2 + bx + c = 0 \) is given by: \[ \Delta = b^2 - 4ac \] For the given equation \( 2x^2 - 5x + k = 0 \), we have: - \( a = 2 \), - \( b = -5 \), - \( c = k \). Substitute these values into the discriminant formula: \[ \Delta = (-5)^2 - 4(2)(k) = 25 - 8k \] For the roots to be real and distinct, we require \( \Delta>0 \): \[ 25 - 8k>0 \] \[ 25>8k \] \[ k<\frac{25}{8} \] Thus, the value of \( k \) must be less than \( \frac{25}{8} \).