# Solving & Graphing Systems of Quadratic Equations

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will define a system of quadratic equations, and it will show how to solve a system of quadratic equations graphically. We will be able to practice the graphing and solving process through a real world application.

Suppose two different businesses can model their profit by the following formulas:

P1 = -15x2 + 8400x - 720000

P2 = -12x2 + 4600x - 50000

where P1 and P2 are the profit for each business, and x is the cost of the products they sell. You work for the company that owns the businesses, and they want you to figure out if there is an equal cost of product that the two businesses could charge in order to have the same profit and what that profit would be.

You let the common profit be P, and plug that in for P1 and P2 in the above formulas to get the following;

P = -15x2 + 8400x - 720000

P = -12x2 + 4600x - 50000

In mathematics, we call this set of equations a system of quadratic equations. A system of quadratic equations is a set of quadratic equations with the same variables.

The solution to a system of quadratic equations is the value of the variables that make all of the equations in the system true. A system of quadratic equations can have one, two, none, or infinitely many solutions.

Well, a solution to our system is exactly what we want, because it would give us values of our variables, P and x that would make both equations true! Those values would give us the equal cost, x, that each business would have to charge in order to have an equal profit, P. Perfect! Let's find out how to do this!

## Solving By Graphing

There are a few different ways to go about solving a system of quadratic equations. First, let's look at solving by graphing. Thankfully, this process is one of the easier ones! An important note when solving by graphing is that a graphing calculator that can find intersection points is absolutely essential for this process.

To solve a system of quadratic equations by graphing, we follow these steps;

1. Graph both of the equations on the same coordinate plane.
2. Find the intersection points of the two graphs. The intersection points are your solutions.

This makes sense, because the intersection points of the graphs of the equations are points that are on both of the graphs, and therefore satisfy both of the equations, which is exactly the definition of the solution of a quadratic system of equations.

Alright! Back to work! Let's take our system of quadratic equations through our steps to find a solution to the problem.

The first step is to graph the two equations on the same graph.

The second step is to find the intersection points of the graphs.

It is probably very clear now why we need to use a graphing calculator that does this for us. I don't know about you, but I certainly can't pinpoint the exact value of the intersection point just by looking at it! Thank goodness for calculators in instances like these! We push a few buttons and wa-la! We have our intersection point!

We see that the intersection point that makes sense with the problem occurs at (211.69631, 386019.09) Great! Rounding these values tells us that if both of the businesses charge \$211.70 for their product, then they will have an equal profit of \$386,019.09.

## Solving Algebraically

Of course, if you don't have a graphing utility on hand to do this for you, you can solve the system algebraically. To solve this system algebraically, we follow these steps;

1. Solve both equations for y, or in this case, P. In our example, this step is done for us.

2. Set the two expressions that you found for y in step 1 equal to each other, and solve for x. In our example, we have the following;

-15x2 + 8400x - 720000 = -12x2 + 4600x - 50000

Solving this equation for x, and rounding, gives x = 211.69631 or x = 1054.970355.

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