Single correct option
If the roots of x² − 5x + k = 0 differ by 3, what is the value of k?
- A. 2
- B. 3
- C. 4
- D. 6
Show answer and solution
Answer: (C) 4
For roots α and β: α + β = 5 and αβ = k.
(α − β)² = (α + β)² − 4αβ = 25 − 4k.
We need (α − β)² = 9, so 25 − 4k = 9 and k = 4.
Check: x² − 5x + 4 = (x − 1)(x − 4), with roots 1 and 4 differing by 3. ✓
Tip: The difference of the roots is √D/|a|. Here √(25 − 4k) = 3.
(α − β)² = (α + β)² − 4αβ = 25 − 4k.
We need (α − β)² = 9, so 25 − 4k = 9 and k = 4.
Check: x² − 5x + 4 = (x − 1)(x − 4), with roots 1 and 4 differing by 3. ✓
Tip: The difference of the roots is √D/|a|. Here √(25 − 4k) = 3.
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