Question 1160754


{{{z^4+2z^3-9z^2-10z+50=0}}} factor

{{{z^4+ 6z^3 +10z^2 -4z^3-24z^2-40z +5z^2+30z +50=0}}} group

{{{(z^4-4z^3+5z^2)+ (6z^3-24z^2+30z)+(10z^2-40z +50)=0}}}

{{{z^2(z^2-4z+5)+ 6z(z^2-4z+5)+10(z^2-4z +5)=0}}}

{{{(z^2 - 4 z + 5) (z^2 + 6 z + 10) = 0}}}

use quadratic function to find roots

for {{{(z^2 - 4 z + 5)=0}}}

{{{z=(-(-4)+-sqrt((-4)^2-4*1*5))/(2*1)}}}

{{{z=(4+-sqrt(16-20))/2}}}

{{{z=(4+-sqrt(-4))/2}}}

{{{z=(4+-2i)/2}}}

{{{z=(2+-i)}}}

roots are: 

{{{z=2+i}}} ->is a root of {{{z^4+2z^3-9z^2-10z+50=0}}}

since complex roots always come in pairs, you also have {{{z=2-i}}}


and remaining roots are:

 
 for{{{ (z^2 + 6 z + 10) = 0}}}

{{{z=(-6+-sqrt(6^2-4*1*10))/(2*1)}}}

{{{z=(-6+-sqrt(36-40))/2}}}

{{{z=(-6+-sqrt(-4))/2}}}

{{{z=(-6+-2i)/2}}}

{{{z=(-3+-i)}}}

roots are: 

{{{z=-3+i}}} 
{{{z=-3-i}}}