### Quartic Equation by Hyperbolic method

 Quartic Equation Solution of quartic equation in hyperbolic form is -R(2 cosh k + el) . For quartic equations: if the roots are a,b,c,d. Then, Re-l = -(Rek + Re-k + Rel) Re-l = -R(2 cosh k + el) (Re-l)4 = [-R(2 cosh k + el)]4 = [-R(2 cosh k + el)]2[-R(2 cosh k + el)]2 = R2[4 cosh2 k + e2l + 4cosh k el][-R(2 cosh k + el)]2 = R2(4cosh k el)[-R(2 cosh k + el)]2 + R2[4 cosh2 k + e2l][-R(2 cosh k + el)]2 = R2(4cosh k el)[-R(2 cosh k + el)]2 + R2[4 cosh2 k + e2l][-R(2 cosh k + el)][-R(2 cosh k + el)] = R2(4cosh k el)[-R(2 cosh k + el)]2 - R3[8 cosh3 k + e3l + 4 cosh2 k el + 2 cosh k e2l][-R(2 cosh k + el)] = R2(4cosh k el)[-R(2 cosh k + el)]2 - R3[8 cosh3 k + e3l + (2cosh k el) (2cosh k + el)][-R(2 cosh k + el)] = R2(6cosh k el)[-R(2 cosh k + el)]2 - R3[8 cosh3 k + e3l][-R(2 cosh k + el)] = R2(6cosh k el)[-R(2 cosh k + el)]2+ R4[16 cosh4 k + e4l + 2 cosh k e3l + 8 cosh3 k el] = R2(6cosh k el)[-R(2 cosh k + el)]2+ R4[16 cosh4 k + e4l + (2 cosh k el) ( 4 cosh2 k + e2l)] = R2(6cosh k el)[-R(2 cosh k + el)]2+ R4[16 cosh4 k + e4l + (2 cosh k el) {( 2cosh2 k + e2l)2 - 2cosh k el)}] = R2(8cosh k el)[-R(2 cosh k + el)]2+ R4[16 cosh4 k + e4l - 2cosh2 k e2l] solve this equation by equating it to x4 = -cx2 -e. Where x4 = -cx2 -e is reduced form by transformation of x4 + b'x3 + c'x2 + d'x + e = 0