11.7: Equations with Radicals on Both Sides
What if you had a radical equation like with a radical sign on both sides? How could you find the solutions to this equation? After completing this Concept, you'll be able to solve radical equations like this one.
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CK-12 Foundation: Special Cases with Radical Equations
Guidance
Often equations have more than one radical expression. The strategy in this case is to start by isolating the most complicated radical expression and raise the equation to the appropriate power. We then repeat the process until all radical signs are eliminated.
Example A
Find the real roots of the equation .
Solution
Check: . The solution checks out.
The solution checks out.
The equation has two solutions: and .
Identify Extraneous Solutions to Radical Equations
We saw in Example 3 that some of the solutions that we find by solving radical equations do not check out when we substitute (or “plug in”) those solutions back into the original radical equation. These are called extraneous solutions. It is very important to check the answers we obtain by plugging them back into the original equation, so we can tell which of them are real solutions.
Example B
Find the real solutions of the equation .
Solution
Check: The solution does not check out.
The equation has no real solutions. is an extraneous solution.
Applications using Special Case of Radical Equations
Radical equations often appear in problems involving areas and volumes of objects.
Example C
Anita’s square vegetable garden is 21 square feet larger than Fred’s square vegetable garden. Anita and Fred decide to pool their money together and buy the same kind of fencing for their gardens. If they need 84 feet of fencing, what is the size of each garden?
Solution
Make a sketch:
Define variables: Let Fred’s area be ; then Anita’s area is .
Find an equation:
Side length of Fred’s garden is
Side length of Anita’s garden is
The amount of fencing is equal to the combined perimeters of the two squares:
Solve the equation:
Check: . The solution checks out.
Fred’s garden is and Anita’s garden is .
Watch this video for help with the Examples above.
CK-12 Foundation: Special Cases with Radical Equations
Vocabulary
- The symbol for a square root is . This symbol is also called the radical sign.
Guided Practice
Find the real solutions of the equation .
Solution
Check: First check :
The solution does not check out.
Then check :
The solution checks out.
The equation has one real solution. is an extraneous solution.
Practice
Find the solution to each of the following radical equations. Identify extraneous solutions.
Extraneous Solution
An extraneous solution is a solution of a simplified version of an original equation that, when checked in the original equation, is not actually a solution.Radical Expression
A radical expression is an expression with numbers, operations and radicals in it.Image Attributions
Description
Learning Objectives
Here you'll learn how to find the real roots and identify the extraneous solutions of radical equations that have radicals on both sides of the equal sign. You'll also solve real-world applications that involve such equations.