Question

If the roots of the quadratic equation \( x^2 - 6x + k = 0 \) are real and distinct, what is the range of values for \( k\)?

• A. \( k>9 \)
• B. \( k<9 \)
• C. \( k>0 \)
• D. \( k<0 \)

Hint:

The discriminant determines the nature of roots. For distinct real roots, ensure \( \Delta>0 \) by comparing it with zero.

Answer 1

The Correct Option is B

Step 1: A quadratic equation \( ax^2 + bx + c = 0 \) has real and distinct roots if its discriminant \( \Delta = b^2 - 4ac \) is greater than 0, i.e., \( \Delta>0 \).
Step 2: For the equation \( x^2 - 6x + k = 0 \), the coefficients are:
- \( a = 1 \),
- \( b = -6 \),
- \( c = k \).
Step 3: Calculate the discriminant: \[ \Delta = (-6)^2 - 4 \cdot 1 \cdot k = 36 - 4k. \]
Step 4: Apply the condition for real and distinct roots: \[ 36 - 4k>0. \]
Step 5: Solve the inequality for \( k \): \[ 36>4k \quad \Rightarrow \quad k<\frac{36}{4} \quad \Rightarrow \quad k<9. \]