If $a + b = 10$ and $ab = 16$, find the value of $a^2 – ab + b^2$ and $a^2 + ab + b^2$.

Given: 

$a + b = 10$ and $ab = 16$

To do: 

We have to find the value of $a^2 – ab + b^2$ and $a^2 + ab + b^2$.

Solution: 

$a + b = 10$

Squaring both sides, we get,

$(a + b)^2 = (10)^2$

$a^2 + b^2 + 2ab = 100$

$a^2 + b^2 + 2 \times 16 = 100$

$a^2 + b^2 + 32 = 100$

$a^2 + b^2 = 100 - 32 = 68$

Therefore,

$a^2 - ab + b^2 = a^2 + b^2 - ab$

$= 68 - 16$

$= 52$

$a^2 + ab + b^2 = a^2 + b^2 + ab$

$= 68 + 16$

$= 84$

Hence, $a^2 - ab + b^2 = 52$ and $a^2 + ab + b^2 =84$.

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Updated on: 10-Oct-2022

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