Equations: FAQ

Rational equations are equations that contain one or more fractions with a variable in the denominator. To solve them, we usually try to get rid of the fractions, and then solve the equation with other strategies we know.

Here are a couple of examples of different rational equations:

Square-root equations are equations that contain square roots. To solve them, we usually isolate the square root on one side of the equation, and square both sides in order to get rid of the square root.

Here are a few examples of different square-root equations:

Sometimes when we are solving an equation, we perform non-reversible operations such as squaring on both sides of the equation. When we do that, we might get a solution that makes one side of the new equation equal to the other, but doesn't actually work when we plug it back into the original equation. This is solution is called an extraneous solution. It's really important to always check our solutions by substituting them back into the original equation, to make sure they're not extraneous.

Cube-root equations are equations that contain cube roots. To solve them, we usually isolate the cube root on one side of the equation, and cube both sides in order to get rid of the cube root.

Quadratic systems are systems of equations where at least one of the equations is a quadratic equation. To solve them, we often use substitution or elimination, just like with linear systems, but sometimes we have to use more advanced techniques like graphing or completing the square.

One way to solve an equation is to graph both sides of the equation as separate functions. Each place where the two graphs intersect is a solution to the equation.

There are countless ways that solving equations comes up in the real world. For example, engineers might use quadratic systems to model the motion of a projectile, or a business analyst might use rational equations to calculate the most profitable price point for a product.