If a,b and c are the roots of x³+qx+r=0, then (a-b)²+(b-c)²+(c-a)²=

Correct Answer is : −6q-6q−6q

Correct Answer is : −6q-6q−6q

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AP EAMCET Engineering 2018 Apr 22 Shift 1 Paper

Section: Mathematics

Question : 13 of 160

Marks: +1, -0

a,b

c

x^3+qx+r=0,

(a-b)^2+(b-c)^2+(c-a)^2=

Solution:

The equation is given as.

x^{3}+q x+r=0

The roots of the given equation is.

a+b+c=0

a b+b c+c a=q

a b c=-r

Squaring the root of the equation.

(a+b+c)^{2}=0

a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a=0

a^{2}+b^{2}+c^{2}+2(a b+b c+c a)=0

Substituting the values we get,

a^{2}+b^{2}+c^{2}=-2 q

Now,

(a-b)^{2}+(b-c)^{2}+(c-a)^{2}

=a^{2}+b^{2}-2 a b+b^{2}+c^{2}-2 b c

+c^{2}+a^{2}-2 a c

=2 (a^{2}+b^{2}+c^{2} )-2(a b+b c+c a)

=9(-2 q)-2(q)

=-6 q

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