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



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