Question 926377
{{{x^2+2xy-y^2=-17}}}..........eq.1
{{{ x^2-y^2=-45}}}..........eq.2
_____________________________________subtract eq.2 from eq.1

{{{x^2+2xy-y^2-(x^2-y^2)=-17-(-45)}}}

{{{cross(x^2)+2xy-cross(y^2)-cross(x^2)+cross(y^2)=-17+45}}}

{{{2xy =28}}}

{{{xy =14}}}.....solve for {{{x}}}

{{{x =14/y}}}.....eq.1a

go to {{{ x^2-y^2=-45}}}..........eq.2, substitute {{{x =14/y}}}


{{{ (14/y)^2-y^2=-45}}}....solve for {{{y}}}


{{{ 14^2/y^2-y^2=-45}}}

{{{ 196/y^2=y^2-45}}}

{{{196=y^2(y^2-45)}}}

{{{196=y^4-45y^2}}}

{{{0=y^4-45y^2-196}}}

{{{y^4+2y^2-49y^2-196=0}}}

{{{(y^4+2y^2)-(49y^2+196)=0}}}

{{{y^2(y^4+2)-49(y^2+4)=0}}}

{{{(y^2-49)(y^2+4)=0}}}

{{{(y^2-7^2)(y^2+4)=0}}}

{{{(y-7)(y+7)(y^2+4)=0}}}

solutions:

if {{{y-7=0}}} =>{{{y=7}}}

if {{{y+7=0}}}=>{{{y=-7}}}

if {{{y^2+4=0}}}=>{{{y^2=-4}}}=>{{{y=sqrt(-4)}}}=>{{{y=2i}}} or {{{y=-2i}}};  complex solutions


so, {{{x =14/y}}} will be

real solutions:

{{{x =14/7}}}=> {{{x =2}}} if {{{y=7}}}
and
{{{x =14/-7}}}=> {{{x =-2}}} if {{{y=-7}}}

complex solutions: 

{{{x =14/2i}}}=> {{{x =7i}}} if {{{y=2i}}}
and
{{{x =14/-2i}}}=> {{{x =-7i}}} if {{{y=-2i}}}


{{{drawing(600, 600, -10, 10, -10, 10,locate(-2,-7,p(-2,-7)),locate(2,7,p(2,7)),circle(-2,-7,.125),circle(2,7,.125), graph(600, 600, -10, 10, -10, 10,sqrt(x^2+45),-sqrt(x^2+45),x+sqrt(2x^2+17) ,x-sqrt(2x^2+17))) }}}