Math Problem Solver

The system of equations has \(2\) solutions.

\[x = 1 - 2 \sqrt{3}, y = -2 + 8 \sqrt{3}\]\[x = 1 + 2 \sqrt{3}, y = - 8 \sqrt{3} - 2\]

Solve \(y = 2 - 4 x\) for \(x\). \[x = \frac{1}{2} - \frac{y}{4}\]Substitute \(\frac{1}{2} - \frac{y}{4}\) for \(x\) in \(3 x^{2} + x y + 11 = 0\) and simplify. $$\begin{aligned}3 x^{2} + x y + 11 &= 0 \\ 3 \left(\frac{1}{2} - \frac{y}{4}\right)^{2} + \left(\frac{1}{2} - \frac{y}{4}\right) y + 11 &= 0 \\ y^{2} + 4 y &= 188 \\y^{2} + 4 y - 188 &= 0 \\ y &= \frac{-(4) \pm \sqrt{(4)^{2} - 4(1)(-188)}}{2(4)} \\ y = -2 + 8 \sqrt{3}&, y = - 8 \sqrt{3} - 2\end{aligned}$$Substitute \(-2 + 8 \sqrt{3}\) into \(y = 2 - 4 x\) to solve for \(x\). \[\begin{aligned}-2 + 8 \sqrt{3} &= 2 - 4 x\\4 x &= 4 - 8 \sqrt{3}\\x &= 1 - 2 \sqrt{3}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = 1 - 2 \sqrt{3},\,y = -2 + 8 \sqrt{3}\end{aligned}$$Substitute \(- 8 \sqrt{3} - 2\) into \(y = 2 - 4 x\) to solve for \(x\). \[\begin{aligned}- 8 \sqrt{3} - 2 &= 2 - 4 x\\4 x &= 4 + 8 \sqrt{3}\\x &= 1 + 2 \sqrt{3}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = 1 + 2 \sqrt{3},\,y = - 8 \sqrt{3} - 2\end{aligned}$$