Question

alpha beta are roots of y square - 2y - 7 is equal to zero then find first Alpha square plus beta squarebeta

Answer 1

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Answer 2

$Given \:\alpha \:and \beta \:are \: roots \:of \\quadratic \:equation \:y^{2} - 2y - 7 = 0$

$Now, Compare \: above \: equation \: with \\ay^{2} + by + c = 0, we \:get$

$a = 1, \: b = -2 \: and\: c = -7$

$i ) Sum \: of \:the \:roots = \frac{-b}{a}$

$\implies \alpha + \beta = \frac{-(-2)}{1}$

$\implies \alpha + \beta = 2 \: ---(1)$

$ii ) Product \: of \:the \:roots = \frac{c}{a}$

$\implies \alpha \beta = \frac{-7}{1}$

$\implies \alpha \beta = -7\: ---(2)$

$iii ) \alpha^{2} + \beta^{2} \\= ( \alpha + \beta )^{2} - 2\alpha \beta\\= 2^{2} - 2 \times (-7) \\= 4 + 14 \\= 18 \: --(3)$

$iv) \alpha^{3} + \beta^{3} \\= ( \alpha + \beta )^{3} - 3 \alpha \beta ( \alpha + \beta )\\= 2^{3}- 3 \times (-7) \times 2 \\= 8 + 42\\= 50 \: --(4)$

Therefore.,

$\red{\alpha^{2} + \beta^{2} }\green {= 18}$

$\red{\alpha^{3} + \beta^{3} }\green {= 50}$

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