Question

if zeros of polynomial x^ 3 - 12 x^2+ 39 x + K are in Arithmetic progression find value of k​

Answer 1

\left[x _{1}\right] = \left[ 4+\frac{\sqrt[3]{\left( -378+\frac{\left( -27\right) \,K}{2}+\frac{\sqrt{\left( 492804+40824\,K+729\,K^{2}\right) }}{2}\right) }}{3}+\frac{9\,\sqrt[3]{2}}{\sqrt[3]{\left( -756 - 27\,K+\sqrt{\left( 492804+40824\,K+729\,K^{2}\right) }\right) }}\right][x​1​​]=​⎣​⎢​⎢​⎢​⎢​⎡​​4+​3​​​3​​√​(−378+​2​​(−27)K​​+​2​​√​(492804+40824K+729K​2​​)​​​​​)​​​​​+​​3​​√​(−756−27K+√​(492804+40824K+729K​2​​)​​​)​​​​​9​3​​√​2​​​​​​⎦​⎥​⎥​⎥​⎥​⎤​​

Arithmetic progression find value