Question 1206952
 {{{a^3 - b^2 = 11}}}.....eq.1
{{{a^2 + b = 13}}}....eq.2
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start with

{{{a^2 + b = 13}}}....eq.2, solve for {{{b}}}

  {{{b = 13-a^2}}}....eq.1a


go to eq.2, substitute {{{b}}}


{{{ a^3 - (13-a^2)^2 = 11}}}.....eq.1

 {{{a^3 - (a^4 - 26 a^2 + 169) = 11}}}

{{{ a^3 - a^4 + 26 a^2 - 169 = 11}}}

 {{{a^3 - a^4 + 26a^2 - 169 -11=0}}}

{{{a^3 - a^4 + 26a^2 - 180=0}}}....factor

{{{-(a - 3) (a^3 + 2 a^2 - 20 a - 60) = 0}}}


real solutions:

{{{-(a - 3)  = 0}}}=> {{{a=3}}}
 
{{{(a^3 + 2 a^2 - 20 a - 60) = 0}}}, using calculator we get {{{a}}}≈{{{4.79}}}

go to

{{{b = 13-a^2}}}....eq.1a, substitute {{{a}}}

{{{b = 13-3^2}}}

{{{b=4}}}


{{{b = 13-4.79^2}}}

{{{b=13-22.9441}}}

{{{b}}}≈{{{-9.94}}}

complex solutions: using calculator

{{{a}}}≈{{{-3.3951 + 0.9996i}}}, {{{b}}}≈{{{2.4727 + 6.7875i}}}
{{{a}}}≈{{{-3.3951 - 0.9996i}}}, {{{b}}}≈{{{2.4727 - 6.7875i}}}


answer:

{{{a=3}}},{{{b=4}}}

{{{a}}}≈{{{4.79}}},{{{b}}}≈{{{-9.94}}}

{{{a}}}≈{{{-3.3951 + 0.9996i}}}, {{{b}}}≈{{{2.4727 + 6.7875i}}}
{{{a}}}≈{{{-3.3951 - 0.9996i}}}, {{{b}}}≈{{{2.4727 - 6.7875i}}}

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