If a+b=2 and a4+b4=272 then a quadratic equation whose roots are a and b is

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If $a + b = 2$ and $a^{4} + b^{4} = 272$ then a quadratic equation whose roots are a and b is

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$x^{2} - 2 x + 16 = 0$

$x^{2} - 2 x + 8 = 0$

$x^{2} - 2 x - 16 = 0$

$x^{2} - 2 x + 32 = 0$

answer is A.

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Detailed Solution

We have,

$a + b = 2$ and $a^{4} + b^{4} = 272$

Now, $a^{4} + b^{4} = 272$

$\begin{array}{l} \Rightarrow {(a^{2} + {\dot{b}}^{2})}^{2} - 2 a^{2} b^{2} = 272 \\ \Rightarrow {\{(a + b)^{2} - 2 a b\}}^{2} - 2 a^{2} b^{2} = 272 \\ \Rightarrow (4 - 2 a b)^{2} - 2 a^{2} b^{2} = 272 \\ \Rightarrow 16 - 16 a b + 2 a^{2} b^{2} = 272 \\ \Rightarrow (a b)^{2} - 8 (a b) - 128 = 0 \\ \Rightarrow (a b - 16) (a b + 8) = 0 \Rightarrow a b = 16 or, - 8 \end{array}$