Solution of two quadratic equation with a particular structure

Asked Feb 15 '20 at 13:43

Active Feb 15 '20 at 14:46

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I have a system of two quadratic equations with unknowns $x$ and $y$ and the following structure

\begin{cases} 0=ax^2-bxy+kx-r_1y+c_1\\ 0=by^2-axy-kx-r_2y+c_2 \end{cases}

where all parameters are nonzero positive scalars, except $a$ and $c_2$ that could also be negative.

Is there a special algebraic solution for the above system? Is there any possibility to get a unique solution?

edited Jun 12 '20 at 10:38

Community

asked Feb 15 '20 at 13:43

Daniele

So $c_2$ is nonzero or can it be zero? – coffeemath Feb 15 '20 at 13:48
all parameters are non-zero – Daniele Feb 15 '20 at 13:57
This is equivalent to cubic equation (no $y^2$ term) so you may have one or three solutions in real numbers. Algebraic solutions of the cubic can be obatained using Cardano's formulas. So at least one solution should exist. – Vasili Feb 15 '20 at 14:04

Solution of two quadratic equation with a particular structure

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