Write the number of real roots of the equation $(x - 1)^{2} + (x - 2)^{2} + (x - 3)^{2} = 0$ .

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Solution

$(x - 1)^{2} + (x - 2)^{2} + (x - 3)^{2} = 0 \Rightarrow x^{2} + 1 - 2 x + x^{2} + 4 - 4 x + x^{2} + 9 - 6 x = 0 \Rightarrow 3 x^{2} - 12 x + 14 = 0$
Comparing the given equation with the general form of the quadratic equation $a x^{2} + b x + c = 0$ , we get $a = 3, b = - 12$ and $c = 14 .$

$D = b^{2} - 4 a c = {(- 12)}^{2} - 4 \times 3 \times 14 = 144 - 168 = - 24 Since the value of D is less than 0, the given equation has no real roots .$

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