Question

The roots of the quadratic equation x² - 5x + k = 0 are real and distinct. What is the range of values for k?

• A. \( k<\frac{25}{4} \)
• B. \( k>\frac{25}{4} \)
• C. \( k \leq \frac{25}{4} \)
• D. \( k \geq \frac{25}{4} \)

Hint:

For quadratic equations, use the discriminant: \( \Delta>0 \) for real and distinct roots, \( \Delta = 0 \) for equal roots.

Answer 1

The Correct Option is A

To find the range of \( k \) for real and distinct roots, we proceed as follows:

1. For a quadratic equation \( ax^2 + bx + c = 0 \), the roots are real and distinct if the discriminant \( \Delta>0 \).
2. Given: \( x^2 - 5x + k = 0 \), so \( a = 1 \), \( b = -5 \), \( c = k \).
3. Calculate the discriminant: \[ \Delta = b^2 - 4ac = (-5)^2 - 4 \times 1 \times k = 25 - 4k. \]
4. Set the condition for real and distinct roots: \[ 25 - 4k>0. \]
5. Solve the inequality: \[ 25>4k \implies 4k<25 \implies k<\frac{25}{4}. \]
6. Match with options: \( k<\frac{25}{4} \) corresponds to option (A).

Thus, the correct answer is: \[ \boxed{k<\frac{25}{4}} \]