Toán Lớp 9: Giải hệ pt : x ³ +3x=y ³ -8 x ²+y ² =y+2

Question

Toán Lớp 9:  Giải hệ pt :   x ³ +3x=y ³ -8                       x ²+y ² =y+2

lanminhngoc · Answer

$ begin{array}{l}  left {  begin{array}{l} {x^3} + 3x = {y^3} - 8 left( 1  right)   {x^2} + {y^2} = y + 2 left( 2  right)  end{array}  right.    left( 1  right) + 3. left( 2  right)     Rightarrow {x^3} + 3{x^2} + 3x + 3{y^2} = {y^3} + 3y - 1 - 1     Leftrightarrow  left( {{x^3} + 3{x^2} + 3x + 1}  right) = {y^3} - 3{y^2} + 3y - 1     Leftrightarrow { left( {x + 1}  right)^3} = { left( {y - 1}  right)^3}     Leftrightarrow x + 1 = y - 1     Leftrightarrow x - y =  - 2  Rightarrow x = y - 2 left( 3  right)    left( 2  right)  Rightarrow {x^2} + {y^2} = y + 2     Leftrightarrow { left( {y - 2}  right)^2} + {y^2} = y + 2     Leftrightarrow 2{y^2} - 4y + 4 = y + 2     Leftrightarrow 2{y^2} - 5y + 2 = 0     Leftrightarrow  left( {y - 2}  right) left( {2y - 1}  right) = 0     Leftrightarrow  left[  begin{array}{l} y = 2  Rightarrow x = 0   y =  dfrac{1}{2}  Rightarrow x =  -  dfrac{3}{2}  end{array}  right.     Rightarrow  left( {x;y}  right) =  left( {0;2}  right), left( { -  dfrac{3}{2}; dfrac{1}{2}}  right)    end{array}$