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A radical equation is an equation which involves one or more variables under radical signs. For instance, √x = 5 is a radical equation since the variable x is found under the square root sign.
Radical equations might appear complex at the very first sight, however, they can be solved using just a few steps. In this article, you will learn what is a radical equation, how radical equations operate, and the general process of solving radical equations.
What Is a Radical Equation?
A radical equation is an equation that contains a variable within a radical symbol.
The radical is a mathematical notation representing a root. The most common type of radical is the square root, denoted by √.
Example: √x = 6
Here, we have a radical equation because x is under the square root symbol.
An example of another radical equation would be: √(x + 4) = 8
In this example, the expression x + 4 is under the radical.
Radical equations may include square roots, cube roots, fourth roots, and many other kinds of roots.
What Is a Radical Equation Example?
Let’s look at some simple examples:
- √x = 7
- √(x + 3) = 5
- 2√x = 10
- √(2x + 1) = 9
- ∛x = 4
In each example, the variable is part of a radical expression.
For example:
√x = 7
To find x, square both sides:
(√x)² = 7²
So:
x = 49
Therefore, the solution is x = 49.
How to Solve a Radical Equation
Solving a radical equation involves a series of four simple steps:
- Isolate the radical.
- Square both sides.
- Solve the new equation.
- Verify the result in the original equation.
Now let us see each step in action through an example.
Step 1: Isolate the Radical
Consider:
√(x + 5) = 9
The radical is already isolated on the left side, so we can move to the next step.
Step 2: Square Both Sides
Because this is a square root, square both sides:
(√(x + 5))² = 9²
This gives:
x + 5 = 81
Step 3: Solve the Equation
Subtract 5 from both sides:
x = 76
So, our possible answer is:
x = 76
Step 4: Check the Answer
Substitute 76 into the original equation:
√(76 + 5) = 9
√81 = 9
9 = 9
The answer is correct.
Therefore:
x = 76
Recommended Reading: How Do You Spell 20?
Why Should You Verify Your Answer in the Radical Equation?
Verification of your answer is an essential process in finding a radical equation solution.
While squaring both sides of the equation, you may come across a solution that cannot be plugged in the original equation. The solution of such a type is termed an extraneous solution.
Therefore, always plug your solution back to the original equation and not the new one which you found out after squaring.
What Is an Extraneous Solution?
An extraneous solution is the solution which satisfies the new equation, but does not satisfy the original equation.
For instance, while doing operations with a radical equation, you may find a potential solution for x. However, before accepting the answer, plug the solution into the original equation.
In case both sides of the equation are equal, the solution is correct.
If not, then the solution is extraneous.
Radical Equation vs. Radical Expression
Radical Equation vs. Radical Expression
A radical expression and a radical equation are related, but they are not the same thing. The key difference is whether the expression includes an equal sign.
| Radical Expression | Radical Equation |
|---|---|
| √x | √x = 5 |
| √(x + 3) | √(x + 3) = 7 |
| ∛x | ∛x = 4 |
The main difference is the equal sign.
This is a radical expression because it does not contain an equal sign.
This is a radical equation because it contains an equal sign and can be solved for a variable.
Common Mistakes When Solving Radical Equations
There are some errors that students commit when working with radical equations.
1. Failing to Isolate the Radical
In an attempt to raise each side to a certain power, make sure to isolate the radical first.
2. Squaring One Side of the Equation
In squaring one side of an equation, ensure that you also square the other side.
For example,
√x = 6
equals to
x = 36
and not x = 6² on one side only.
3. Not Checking the Solution
Always verify the solution in the original radical equation.
4. Failing to Recognize Extraneous Solutions
Not all values obtained through the process are solutions to the original equation.
Conclusion
A radical equation is an equation that has a variable within the radical symbol. Squareroots are the most commonly seen, but they can also have cube roots and other roots.
Solving radical equations requires you to follow the basic steps, which include isolation of the radical, raising both sides of the equation to powers, solving the equation, and finally checking the answer. In case the equation gets too difficult to handle, there is a radical equation calculator that will help.
Through consistent practice, the task becomes very easy. Begin by working on simple squareroot equations then move to complex ones.
You can opt for our Advanced Math or Vedic Math+Mental Math courses. Our Math Quiz for grades 3rd, 4th, 5th, and 6th helps in further exciting and engaging in mathematics with hands-on lessons.
