Numerical answer
The roots of the equation x² − 7x + k = 0 differ by 3. What is the value of k?
Show answer and solution
Answer: 10
Let the roots be α and β. Then α + β = 7 and αβ = k.
(α − β)² = (α + β)² − 4αβ ⇒ 9 = 49 − 4k ⇒ k = 10.
Check: x² − 7x + 10 = (x − 2)(x − 5); roots 2 and 5 differ by 3 ✓.
Tip: For any quadratic ax² + bx + c, the difference of roots is √(b² − 4ac)/|a|.
(α − β)² = (α + β)² − 4αβ ⇒ 9 = 49 − 4k ⇒ k = 10.
Check: x² − 7x + 10 = (x − 2)(x − 5); roots 2 and 5 differ by 3 ✓.
Tip: For any quadratic ax² + bx + c, the difference of roots is √(b² − 4ac)/|a|.
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