If $a>b>c$ and $a^3+b^3+c^3=3abc,$ then the quadratic equations $ax^2+bx+c=0$ has roots which are

Using the condition for real roots of a quadratic equation, we evaluate the discriminant D equals b squared minus 4ac. The identity a cubed plus b cubed plus c cubed minus 3abc factors to (a+b+c) times half of ((a-b) squared plus (b-c) squared plus (c-a) squared). Since a, b, and c are distinct real numbers, the only way this expression equals zero is if a+b+c equals 0. Substituting c equals -(a+b) into the discriminant gives b squared minus 4a(-(a+b)), which simplifies to b squared plus 4a squared plus 4ab, or (2a+b) squared, a positive value; the roots are both real.

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