Weird Equations

Algebra Level 5

x 4 + 2 y 3 − x = − 1 4 + 3 3 y 4 + 2 x 3 − y = − 1 4 − 3 3 \begin{aligned} x^4+2y^3 - x & = -\frac{1}{4} + 3\sqrt{3} \\ y^4 + 2x^3 - y & = -\frac{1}{4} - 3\sqrt{3} \end{aligned}

Find the number of solutions in real numbers to the system of equations above.

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Alan Yan by Alan Yan , 20, USA

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1 solution

Alan Yan
Nov 11, 2015

Add the equations and you get ( x 4 + 2 x 3 − x + 1 4 ) + ( y 4 + 2 y 3 − y + 1 4 ) = 0 ( x 2 + x − 1 2 ) 2 + ( y 2 + y − 1 2 ) 2 = 0 ⟹ x = − 1 ± 3 2 , y = − 1 ± 3 2 \begin{aligned} \left(x^4 + 2x^3 - x + \frac{1}{4}\right) + \left(y^4 + 2y^3 - y + \frac{1}{4}\right) & = 0 \\ \left(x^2 + x - \frac{1}{2}\right)^2 + \left(y^2 + y - \frac{1}{2}\right)^2 & = 0 \\ \implies x = \frac{-1 \pm \sqrt{3}}{2} , y = \frac{-1 \pm \sqrt{3}}{2} \end{aligned} We can check that the only solution is ( x , y ) = ( − 1 − 3 2 , − 1 + 3 2 ) (x,y) = \left(\frac{-1 - \sqrt{3}}{2} , \frac{-1 + \sqrt{3}}{2}\right) .

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You should write We check.....only real solution is

Department 8 - 5 years, 7 months ago

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