\begin{array} { l } { \text { If a, b, c are distinct real numbers } } { \text { and } a ^ { 3 } + b ^ { 3 } + c ^ { 3 } = 3 a b c , \text { then the equation } } \ { a x ^ { 2 } + b x + c = 0 \text { has two roots, out of which one } } { \text { root is } } \end{array}

Using the identity a cubed plus b cubed plus c cubed minus 3abc equals (a plus b plus c) multiplied by (a squared plus b squared plus c squared minus ab minus bc minus ca), the given equation a cubed plus b cubed plus c cubed equals 3abc implies that (a plus b plus c) multiplied by (a squared plus b squared plus c squared minus ab minus bc minus ca) equals 0. Since a, b, and c are distinct real numbers, the second term cannot be zero, leaving us with a plus b plus c equal to 0, or c equal to negative a minus b. Substituting c into the quadratic equation ax squared plus bx plus c equals 0 yields ax squared plus bx minus a minus b equals 0, which factors as (x minus 1) multiplied by (ax plus a plus b) equals 0. The roots are 1 and negative (a plus b) divided by a; replacing negative (a plus b) with c gives the second root as c divided by a.

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