If $x^{2} + x + 1 = 0$ and $x^{2} + a x + b = 0$ have a common root, then the minimum value of $(x - a)^{2} + 2 b$ is

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Solution

The correct option is C $2$
$x^{2} + x + 1 = 0$ has non-real roots.
Also, the coefficients are real.
$\Rightarrow$ Both the roots are common.
$∴ a = b = 1$
So, $(x - a)^{2} + 2 b = (x - 1)^{2} + 2$
$∴$ Minimum value $= 2$

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The correct option is C $2$
$x^{2} + x + 1 = 0$ has non-real roots.
Also, the coefficients are real.
$\Rightarrow$ Both the roots are common.
$∴ a = b = 1$
So, $(x - a)^{2} + 2 b = (x - 1)^{2} + 2$
$∴$ Minimum value $= 2$