If a + b = 10 and ab = 16, find the value of $a^{2} - a b + b^{2} a n d a^{2} + a b + b^{2}$ .

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Solution

a + b = 10, ab = 16

Squaring,

$(a + b)^{2} = (10)^{2} \Rightarrow a^{2} + b^{2} + 2 a b = 100 \Rightarrow a^{2} + b^{2} + 2 \times 16 = 100 \Rightarrow a^{2} + b^{2} + 32 = 100 ∴ a^{2} + b^{2} = 100 - 32 = 68 N o w, a^{2} - a b + b^{2} = a^{2} + b^{2} - a b = 68 - 16 = 52 a n d a^{2} + a b + b^{2} = a^{2} + b^{2} + a b = 68 + 16 = 84$

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