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(a-b)^2 Expansion and Visualisation

If the values...

Question

If the values of $(a - b)$ and $a b$ are $6$ and $40$ respectively, find the values of $a^{2} + b^{2}$ and $(a + b)^{2}$

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Solution

$(a - b) = 6$
$\Rightarrow a b = 40$
$\Rightarrow {(a - b)}^{2} = {(a + b)}^{2} - 4 a b$
$\Rightarrow {(a + b)}^{2} = {(a - b)}^{2} + 4 a b$
$= 6^{2} + 4 (40)$
$= 36 + 160$
$= 196.$
$\Rightarrow a^{2} + b^{2} = {(a + b)}^{2} - 2 a b$
$= (196) - 2 (40)$
$= 196 - 80$
$= 116.$
Hence, the answer is $116.$

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