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Solving Radical Equations
Radical equations are equations contain radical expressions. The radical equations we are going to solve are mainly square root equations and cubic root equations.
Example #1: Solve x = 8
Solution: The first thing we need to do to solve radical equations is to remove the radical ( n th roots).
x = 8
To remove the square root on the left side, we will need to square both sides of the equation.
( ) ( )^2
2 x = 8 x = 64
Simplify each side of the equation.
64 = 8 8 = 8 Check the answer. x = 64 is the solution.
Example #2: Solve 2 x โ 5 = 3
Solution:
This equation looks a little different than the previous one. The radicand (the expression under the radical sign) of the previous equation is x. The radicand of this equation is 2 x โ 5. But, as long as the radical term is isolate, we can follow the same steps to solve the equation as mentioned above.
2 x โ 5 = 3
To remove the square root on the left side, we will need to square both sides of the equation.
( ) ( )^2
2 2 x โ 5 = 3 2 x โ 5 = (^92) x = 14 x = 7
Simplify each side of the equation. Solve for x.
Check the answer. x = 7 is the solution.
This instructional aid was prepared by the Tallahassee Community College Learning Commons.
Example #3: Solve 2 x + 8 = x
Solution:
2 x + 8 = x
To remove the square root on the left side, we will need to square both sides of the equation.
( ) ( )^2
2 2 x + 8 = x 2 x + 8 = x^2
Simplify each side of the equation. Solve for x.
x^2 โ 2 x โ 8 = 0
( x โ 4 )( x + 2 ) = 0
x = 4 or x =โ 2
To solve a quadratic equation, we need to set one side of the equation equal to zero. Then factor the equation.
We have to check the solutions to see if they work. When substitute 4 into the equation, we receive a true statement. Therefore 4 is a solution. When substitute -2 into the equation, the result is not a true statement. So -2 is not a solution. x = (^4) is the solution
This instructional aid was prepared by the Tallahassee Community College Learning Commons.
Example #6: Solve 3 x =โ 4
Solution:
The first thing we need to do to solve this radical equation is to remove the radical ( n th roots).
(^3 x )^3 =( โ 4 )^3 x =โ 64
The radical is isolated. We will need to cube both sides of the equation to remove the cubic root on the left side.
(^3) โ 64 =โ 4
We have to check the solution to see if it works. x =โ 64 is the solution.
Example #7: Solve 3 x + 10 = 4
Solution:
(^3) x + 10 = 4 (^3 x + 10 )^3 =( ) 43 x + 10 = 64 x = 54
The radical is isolated. We will need to cube both sides of the equation to remove the cubic root on the left side.
(^3 54) + 10 = (^364) = 4
We have to check the solution to see if it works.
x = 54 is the solution.
Exercises: Solve the following radical equations
1. 3 y โ 1 = 5 2.^3 x โ 4 =โ 2 3. x โ 1 = 5 x โ 9 4. d + 6 = d 5. x โ 1 = x โ 7
Answers:
This instructional aid was prepared by the Tallahassee Community College Learning Commons.
