Single correct option
If the roots of the equation x² − 7x + k = 0 differ by 3, what is the value of k?
- A. 12
- B. 6
- C. 8
- D. 10
Show answer and solution
Answer: (D) 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: the roots are 5 and 2 (sum 7, difference 3, product 10) ✓.
Why others fail: 12 gives roots 3 and 4 (difference 1); 6 gives roots 1 and 6 (difference 5); 8 gives irrational roots with difference √17.
Tip: Whenever a question mentions the difference of roots, reach for (α − β)² = (α + β)² − 4αβ.
(α − β)² = (α + β)² − 4αβ ⇒ 9 = 49 − 4k ⇒ k = 10.
Check: the roots are 5 and 2 (sum 7, difference 3, product 10) ✓.
Why others fail: 12 gives roots 3 and 4 (difference 1); 6 gives roots 1 and 6 (difference 5); 8 gives irrational roots with difference √17.
Tip: Whenever a question mentions the difference of roots, reach for (α − β)² = (α + β)² − 4αβ.
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