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Question
The roots of the equation 2x2 – 7x + 5 = 0 are α and β. Without solving for the roots, find `(alpha + 2)/(beta + 2) + (beta + 2)/(alpha + 2)`
Solution
2x2 – 7x + 5 = `x^2 - 7/2x + 5/2` = 0
α + β = `7/2`
αβ = `5/2`
`(alpha + 2)/(beta + 2) + (beta + 2)/(alpha + 2)`
= `((alpha + 2)^2 + (beta + 2)^2)/(alphabeta + 2alpha + 2beta + 4)`
= `(alpha^2 + 4alpha + 4 + beta^2 + 4beta + 4)/(alphabeta + 2(alpha + beta) + 4)`
= `((alpha + beta)^2 - 2alphabeta + 4(alpha + beta) + 8)/(alphabeta + 2(alpha + beta) + 4)`
= `(49/4 - 10/2 + 28/2 + 16/2)/(5/2 + 14/2 + 8/2)`
= `(49 - 20 + 56 + 32)/(5 + 14 + 8) xx 1/2`
= `117/54`
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