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Question

A

The roots are 2a+b3 and a+2b3

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B

The roots are 2a+b3 and a2b3

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C

The roots are 5a+b3 and a+2b3

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D

The roots are 2a+b3 and a2b4

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Solution

## The correct option is A The roots are 2a+b3 and a+2b3 We have, 9x2−9(a+b)x+(2a2+5ab+2b2)=0 Comparing this equation with Ax2+Bx+C=0, we have A=9,B=−9(a+b) and C=2a2+5ab+2b2 ∴D=B2−4AC ⇒D=81(a+b)2−36(2a2+5ab+2b2) ⇒D=81(a2+b2+2ab)−(72a2+180ab+72b2) ⇒D=9a2+9b2−18ab ⇒D=9(a2+b2−2ab) ⇒D=9(a−b)2≥0 ⇒D≥0 So, the roots of the given equation are real and are given by α=−B+√D2A=9(a+b)+3(a−b)18=12a+6b18=2a+b3 and, β=−B−√D2A=9(a+b)−3(a−b)18=6a+12b18=a+2b3

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