Roots of the equation $x^{2} - 14 x + 38 = 0$ is/are?

11 + 2 \sqrt{7}

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7 - \sqrt{11}

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7 + \sqrt{11}

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11 - 2 \sqrt{7}

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Solution

The correct option is C $7 + \sqrt{11}$
Given: $x^{2} - 14 x + 38 = 0$

Now, comparing the equation with $(a - b)^{2} = a^{2} - 2 a b + b^{2},$
we get $a = x,$ thus the equation becomes:
$(x - b)^{2} = x^{2} - 2 b x + b^{2}$

Now, comparing again, we get $b = 7$
Now, adding $49$ on both sides of the equation, we get the equation as:
$x^{2} - 14 x + 49 - 49 + 38 = 0 \Rightarrow (x - 7)^{2} - 11 = 0 \Rightarrow (x - 7)^{2} = 11 \Rightarrow x - 7 = \pm \sqrt{11} \Rightarrow x = 7 + \sqrt{11} or x = 7 - \sqrt{11}$

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