Write the number of real roots of the equation
Question:

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

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.