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If the roots of the equation (c^(2)-ab)x...

If the roots of the equation `(c^(2)-ab)x^(2)-2(a^(2)-bc)x+b^(2)-ac=0` in x are equal, show that either `a=0` or `a^(3)+b^(3)+c^(3)=3abc`.

Text Solution

Verified by Experts

Since the roots are equal
`:.D=0`
`implies4(b^(2)-ac)^(2)-4(a^(2)-bc)(c^(2)-ab)=0`
`implies(b^(2)-ac)^(2)-(a^(2)-bc)(c^(2)-ab)=0`
`=b(a^(3)-b^(3)+c^(3)-3abc)=0`
`impliesb=0` or `a^(3)+b^(3)+c^(3)-3abc=0`
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