Using the Quadratic Formula to Solve Systems of Non-Linear Equations

In other words, sometimes you'll need to use old tools in new ways when solving the more advanced systems of non-linear equations. The example below demonstrates how the Quadratic Formula is sometimes useful in solving systems; it also shows how involved your computations might get. (Hint: Very.)

• Solve the following system of equations:
   
  x² − xy + y² = 21
  x² + 2xy − 8y² = 0

---

Warning: Do not try to write the solution points as "" or "", because this is not true. Not all combinations of these x-values and y-values are solution points. Don't be sloppy; write the solution out correctly.

By the way, my graph of the system looks like this:

(To graph the ellipse using the traditional methods, you would have to do a "rotation of axes", a process you probably won't see until calculus, if at all.)

---

By the way, there is another method of doing the algebra for the above exercise, because the quadratic in the second equation happens to be factorable. (This factorability is NOT generally true, but you should try to remember to check, just in case.) If you factor the second equation and solve for x in terms of y, you get:

You can plug these expressions for x= into the first equation in place of x, and solve the resulting equation for the corresponding y-values.

This last example (the first way I worked it) is about as complicated as these things ever get. But if you plug away and work neatly and completely, you should be able to arrive at the solution successfully. And if you have a graphing calculator (and the time), try doing a quick graph to verify your answers visually.

You'll hardly ever need to plug an equation into the Quadratic Formula in order to create expressions containing just one of the variables. But when nothing works, see if the Quadratic Formula might help.