If the difference between the corresponding roots of x² + ax + b = 0 and x² +…
2019
If the difference between the corresponding roots of x² + ax + b = 0 and x² + bx + a = 0 is same and a ≠ b, then:
- A.
a + b + 4 = 0
- B.
a + b − 4 = 0
- C.
a − b + 4 = 0
- D.
a − b − 4 = 0
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Correct answer: A
Given: the equations x² + ax + b = 0 and x² + bx + a = 0. Let the roots of the first be r1 and r2, and the roots of the second be s1 and s2.
Sum and product relations: r1 + r2 = −a, r1·r2 = b; and s1 + s2 = −b, s1·s2 = a.
Condition on corresponding roots: the difference between corresponding roots is the same, so r1 − s1 = r2 − s2. This implies (r1 − r2) = (s1 − s2), and squaring gives (r1 − r2)² = (s1 − s2)².
Express differences via sums and products: (r1 − r2)² = (r1 + r2)² − 4r1r2 = a² − 4b, and (s1 − s2)² = b² − 4a. Equating gives a² − 4b = b² − 4a.
Rearrange: a² − b² + 4a − 4b = 0 ⇒ (a − b)(a + b) + 4(a − b) = 0 ⇒ (a − b)(a + b + 4) = 0.
Conclusion: since a ≠ b, the factor (a − b) cannot be zero, so a + b + 4 = 0.