Determine all real solutions to the following system of equations: $$ \begin{cases} y = 4x^3 + 12x^2 + 12x + 3\\ x = 4y^3 + 12y^2 + 12y + 3. \end{cases} $$
Problem
Source: Canada RepĂȘchage 2018/1
Tags: algebra, system of equations
09.04.2018 16:29
$$ \begin{cases} y+1 = 4(x+1)^3\\ x+1 = 4(y+1)^3. \end{cases} $$So $y+1=4^4(y+1)^9 \to y=-1$ or $2^8(y+1)^8 =1 \to 2(y+1)=\pm 1 \to y=-\frac{1}{2},-\frac{3}{2}$ Answer: $(-1;-1),(-\frac{1}{2},-\frac{1}{2});(-\frac{3}{2},-\frac{3}{2})$
10.04.2018 05:20
Real: $(-1,-1)$, $\left(-\frac{3}{2},-\frac{3}{2}\right)$, $\left(-\frac{1}{2},-\frac{1}{2}\right)$ Complex: $\left(-1\pm\frac{i}{2},-1\mp\frac{i}{2}\right)$, $\left(-1+\frac{\sqrt{2}}{4}\pm\frac{\sqrt{2}}{4}i,-1-\frac{\sqrt{2}}{4}\pm\frac{\sqrt{2}}{4}i\right)$, $\left(-1-\frac{\sqrt{2}}{4}\pm\frac{\sqrt{2}}{4}i,-1+\frac{\sqrt{2}}{4}\pm\frac{\sqrt{2}}{4}i\right)$
29.04.2018 01:11
How did you do @rchokler to find the Complex solutions??
29.04.2018 02:33
LittleKesha wrote: How did you do @rchokler to find the Complex solutions?? roots of unity
20.09.2018 15:35
Wolfram Alpha