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Question
If α and β are roots of the quadratic equation x2 – 7x + 10 = 0, find the quadratic equation whose roots are α2 and β2.
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Solution
Given α and β are roots of x2 – 7x + 10 = 0
∴ α + β = `-((-7))/1` = 7 ...(1)
and αβ = `10/1` = 10 ...(2)
∴ (α + β)2 = α2 + β2 + 2αβ
∴ (7)2 = α2 + β2 + 2(10) ...(from (1) and (2))
`\implies` α2 + β2 = 49 – 20 = 29 ...(3)
Now, the quadratic equation whose roots are α2 and β2 will be
x2 – (α2 + β2) = x + α2β2 = 0
`\implies` x2 – 29x + (10)2 = 0 ...(from (2) and (3))
`\implies` x2 – 29x + 100 = 0
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