Single correct option
The roots of the equation x² − 11x + k = 0 differ by 5. What is the value of k?
- A. 24
- B. 28
- C. 30
- D. 18
Show answer and solution
Answer: (A) 24
Let the roots be α and β. Then α + β = 11 and αβ = k.
(α − β)² = (α + β)² − 4αβ ⇒ 25 = 121 − 4k ⇒ k = 24.
Check: x² − 11x + 24 = (x − 3)(x − 8); roots 3 and 8 differ by 5 ✓.
Tip: Alternatively, the roots are (11 ± 5)/2 = 8 and 3 directly, and k is their product.
(α − β)² = (α + β)² − 4αβ ⇒ 25 = 121 − 4k ⇒ k = 24.
Check: x² − 11x + 24 = (x − 3)(x − 8); roots 3 and 8 differ by 5 ✓.
Tip: Alternatively, the roots are (11 ± 5)/2 = 8 and 3 directly, and k is their product.
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