What are "Quadratic" Equations?
Have you ever thrown a ball straight up into the air? Have you ever been on a boat and tried to calculate your speed relevant to the water’s current? Have you ever sold a good or service and wanted to ensure you’d earn a profit while maintaining a competitive price in the market? If you answered “yes” to any of these questions, you’ve already experienced a quadratic equation in real life. And if you haven’t, there’s a good chance you’ll come across one in the future.
Quadratic equations are a special subset* of algebraic equations. This means that while all quadratic equations are algebraic equations, not all algebraic equations are quadratic. So, what makes an equation a quadratic equation?
What Makes an Equation Quadratic?
Luckily for you, quadratic equations are super easy to spot. Does your equation start** with a x-squared ? If so, it’s quadratic! Yes, it’s really that simple. Quadratic equations typically take the following form, where a, b, and c are constants and a is not equal to zero:
But let’s say you’re given an image of a graph without its equation. How can you identify the graph as quadratic? You can think of quadratic equations as “smiley faced” graphs. If your graph is “smiling” or “frowning,” your equation is quadratic. We’ll take a look at some examples in a moment.
Quadratic equations have another special characteristic which is critical to their interpretation. We consider the solutions to quadratic equations the values where y is equal to zero. On a graph, this is shown as the point(s) where the graph crosses the x-axis. Now let’s take a look at some examples.
What Do Quadratic Equations Look Like?
Example 1: One Real Solution
The “smiling” graph above is the visual representation of y = x^2. The solution to this equation, or where y equals zero, only occurs when x also equals zero. Don’t believe me? Try substituting various numeric values for x into the equation above—you’ll only get y exactly equal to zero when x is set to zero.
You may be thinking, “but the graph doesn’t cross the x-axis. You said the solution is where the graph crosses the x-axis.” That is a very valid point! In the example above, the equation does not cross the x-axis, but rather touches the axis in one spot. We still consider this a valid solution, and grant it an exception to the requirement of crossing the axis completely. But, note that in this case, we only have one real solution rather than two.
Example 2: Two Real Solutions
The “frowning” graph above is the visual representation of y = -(x^2) + 16. The solutions to this equation, or where y equals zero, are found where x equals ± 4. You can see on the graph above that the graph crosses the x axis at these values exactly. You can also confirm this algebraically, by setting the equation equal to zero and solving for x.
Example 3: Complex Solutions
The “smiling” graph above is the visual representation of y = (x^2) + 4. The question now is… what are the solutions? The graph doesn’t cross or touch the x-axis, so we can’t find the answer visually. We can try to solve the equation algebraically, but if you set the equation equal to zero and solve for x, you’ll have to take the square root of a negative number. In order to find the square root of a negative number, you’ll need to use the complex number set… and that is a topic for another day!
We just saw a few examples of quadratic graphs and their equations, but there are many more variations than just these three. For an added challenge, think about how quadratic graphs may change as you manipulate their equations:
How do you shift the graph up? Down?
How do you make the graph wider? Narrower?
How do you shift the graph to the left? To the right? (This is trickier!)
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*A subset is a smaller group within a larger group that inherits the characteristics of the larger group while maintaining its own properties.
**Since we order the variables in our equations by the highest power first, this means your equation must have x-squared as the highest power. We’ll address higher order polynomials in a later post.