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How to Solve a Quadratic Equation by Graphing

Instructor: Laura Pennington

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

This lesson will show us how to solve a quadratic equation by graphing. Through definitions, illustrations, and applications, we will learn the steps involved in this process.

Quadratic Equations

Suppose Brian is at the driving range. He hits a golf ball in such a way that its height off the ground can be modeled with the following formula:

Height of the ball = -16x2 + 100x

Where x is the time, in seconds, after he has hit the ball.


solquadeq1


Nice drive! He finds himself curious about how long the ball was in the air (or when the ball hits the ground). Well, let's see here. The ball hits the ground when the height is equal to zero, so if we just set our formula equal to zero, then we have an equation that we can solve to satisfy his curiosity.

-16x2 + 100x = 0

This type of equation is called a quadratic equation. A quadratic equation is a polynomial equation, with highest exponent 2, that can be put in the form ax2 + bx + c = 0, where a, b, and c are constants. There are many ways to solve quadratic equations, but in this lesson, we are going to focus on one method, and that is by graphing.

Solving Quadratic Equations By Graphing

Solving quadratic equations by graphing all revolves around the fact that the intersection points of two graphs are points that satisfy both of the graphed equations. Therefore, if we take a quadratic equation and let y1 equal the left-hand side of the equation and y2 equal the right-hand side of the equation, then when we can graph both of these equations, the x-values of their intersection points will give us the solutions to the original equation.

Let's clarify a bit by breaking this solving method into steps. Given a quadratic equation ax2 + bx + c = d, we take the following steps to solve by graphing:

  1. Let y1 = ax2 + bx + c and y2 = d.
  2. Graph y1 and y2 on the same graph.
  3. Find the intersection points of the two graphs. The x-values of the intersection points are the solutions to your equation.
  4. Check your answer with the original equation.

This doesn't look so bad. Let's take Brian's equation through these steps!

Brian's equation is -16x2 + 100x = 0, so we let

y1 = -16x2 + 100x

y2 = 0

That's easy enough! Next step, we want to graph both equations on the same graph. You can do this by hand, or you can use a graphing calculator. Using a graphing calculator will make step three much easier as we'll see in a minute.


solquadeq2


Still not so bad! Now comes the tricky part! We have to find the intersection points of the two graphs.


solquadeq3


If it's obvious from the graph, we can do this by eye. However, that leaves a lot of room for human error. We could also do it algebraically, but to do that, we would set the two equations equal to each other, getting -16x2 + 100x = 0, and solve for x. But wait! That's where we started, and this lesson has to do with solving by graphing. Our last and best option is to use a graphing calculator. Graphing calculators can graph your equations for you and find their intersection points - definitely the easiest route! Of course, each graphing calculator has different ways to find intersection points, but your calculator manual should explain exactly how to do this!


solquadeq4


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