An equation that contains a radical expression, such as a square root, is called a radical equation. Solving radical equations requires applying the rules of exponents and following some basic algebraic principles. In some cases, it also requires looking out for errors generated by raising unknown quantities to an even power.

Isolate a Radical Term

A basic strategy for solving radical equations is to isolate the radical term first, and then raise both sides of the equation to a power to remove the radical.

There are two key ideas that you will be using to solve radical equations. The first is

if [latex] a=b[/latex], then [latex] {{a}^{2}}={{b}^{2}}[/latex].

This property allows you to square both sides of an equation and remain certain that the two sides are still equal. The second is that if the square root of any nonnegative number x is squared, then you get x.

For [latex]x \geq 0, {{\left( \sqrt{x} \right)}^{2}}=x[/latex].

This property allows you to “remove” the radicals from your equations.

Let’s start with a radical equation that can be solved in a few steps:

Let’s look at another example.

In these examples, the radical term appeared by itself on one side of the equation. If the equation contains additional terms on the side containing the radical, you must begin by isolating the radical. That is, get the radical by itself on one side of the equation.

In the following video, you will see two more examples that are similar to the ones above.

Identify a Radical Equation with No Solutions or Extraneous Solutions

It is important to check your solutions—especially when solving radical equations. Look carefully at the next example.

The answer does not produce a true statement when substituted back into the original equation. Notice that the radical is isolated, we had [latex]\sqrt{x}=-3[/latex]. The principal square root of a number can only be nonnegative. This means that no value for a will result in a radical expression whose positive square root is [latex]-3[/latex].

Incorrect values of the variable, such as those that are introduced as a result of the squaring process, are called extraneous solutions. Extraneous solutions can be identified because they will not create a true statement when substituted back into the original equation. This is one of the reasons why checking your work is so important—if your answers does not satisfy the original equation you need to exclude it from the solution set.

In the following video, we present more examples of solving radical equations by isolating a radical term on one side.