If $m$ and $n$ are the roots of the equation $x^{2} - 5 x + 3 = 0$ , find the values of
i) $(m + n)^{2} + (m - n)^{2}$
ii) $(m + n)^{3} + 4 m n$

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Solution

$x^{2} - 5 x + 3 = 0$
$a = 1, b = - 5, c = 3$
$m + n = \frac{- b}{a} = \frac{- (- 5)}{1} = 5$
$m + n = \frac{c}{a} = \frac{3}{1} = 3$

i) $(m + n)^{2} + (m - n)^{2} = (m + n)^{2} + [(m + n)^{2} - 4 m n]$
$= 5^{2} + 5^{2} - 4 (3)$
$= 25 + 25 - 12$
$= 50 - 12$
$= 38$

ii) $(m + n)^{3} + 4 m n = (5)^{3} + 4 (3)$
$= 125 + 12$
$= 137$ .

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