Question

If the roots of quadratic equation (x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a) are equal then show that a=b=c

Answer 1

Hey mate.
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Given,

(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)

If we multiply out our given equation we get,

3x²-2(a+b+c)x+ab+bc+ca=0

If roots are equal,

B²=4AC

=> 4(a+b+c)²=4*3(ab+bc+ca)

=>(a+b+c)²=3(ab+bc+ca)

=> a²+b²+c²+2(ab+bc+ca)=3(ab+bc+ca)

=> a²+b²+c²=(ab+bc+ca)...(I)

Now,

If b=a , c=a [a=b=c]

(I) becomes ,

a²+(a²)+(a²)=a(a)+(a)(a)+(a)a

WHICH IS TRUE.

∴ASSUMPTION is correct.,

that is a=b=c .

#racks