If the roots of the equation x^2 + p x + q = 0 differ by 1, which relation between p and q is correct?
Answer and explanation
Correct answer: A. p^2 = 4q + 1
- Step 1: Let roots be α and β. Then |α - β| = 1.
- Step 2: (α - β)^2 = (α + β)^2 - 4 α β = 1^2 = 1.
- Step 3: Sum of roots α + β = -p; product α β = q.
- Step 4: (-p)^2 - 4q = 1 => p^2 - 4q = 1 => p^2 = 4q + 1.
Why not the others: p^2 = 4q - 1, p^2 = 2q + 1 and p^2 = 4q + 4 do not fit.
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