1. Solving Radical Equations
Digital Lesson
Solving Radical Equations

2. 1. Show that 12 is a solution of . (12) 
Equations containing variables within radical signs are called radical equations.
A solution to a radical equation is a real number which, when substituted for the variable, gives a true equation.
Examples:
1. Show that 12 is a solution of
(12) 
True.
2. Show that 0 is a solution of
(0)  
True.
3. Show that has no solutions.
The symbol indicates the positive or principal square root of a number.
Since must be positive, has no solutions.
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Radical Equations

3. Property of Raising Each Side of an Equation to a Power
If a and b are real numbers, n is a positive integer, and a = b, then
a n = b n
This principle can be applied to solve radical equations.
Examples: 1. If , then  x = 4.
2. If , then  x = 125.
The converse of the statement is true for odd n and false for even n.
If n is odd and a n = b n then a = b.
If n is even and a n = b n then it is possible that a  b.
Example: (3)2 = 9 = (3)2 but, 3  3.
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Property of Raising Each Side of an Equation to a Power

4. Radical Equations With One Square Root
To solve a radical equation containing one square root:
1. Isolate the radical on one side of the equation.
2. Square both sides of the equation.
3. Solve for the variable.
4. Check the solutions.
Example: Solve
Original equation
Isolate the square root.
Square both sides.
Solve for x.
(22)
Check.
True
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Radical Equations With One Square Root

5. Example: Solve . (1) (9) Original equation Isolate the square root.
Square both sides.
Solve for x.
Solutions
(1)
Check.
True
(9)
Check.
True
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Example: Solve

6. Example: Solve . (9) Original equation Isolate the cube root.
Cube both sides.
Expand the cube.
Simplify.
Group terms.
(x2 + 3) does not give a root because the square root of a negative number is not a real number.
Factor.
Solution
(9)
Check.
True
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Example: Solve

7. Radical Equations With Two Square Roots
To solve a radical equation containing two square roots:
1. Isolate one radical on one side of the equation.
2. Square both sides of the equation.
3. Isolate the other radical.
4. Solve the equation.
5. Check the solutions.
Example: Solve
Isolate one radical.
Square both sides.
Simplify.
Isolate the other radical.
Square both sides.
Solution
(1)
Check.
True
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Radical Equations With Two Square Roots

8. 2 is a solution of the original radical equation.
Example: Solve
Original equation
Isolate the square root.
Square both sides.
Simplify.
Factor.
Possible solutions
(2)
Check 2.
True
2 is a solution of the original radical equation.
(14)
Check 14.
False
14 is not a solution of the original radical equation.
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Example: Solve

9. Let x be the distance from the bottom of the board to the wall.
Example: A ten-foot board leans against an 8-foot wall so that the top end of the board is at the top of the wall How far must the bottom of the board be from the wall?
Let x be the distance from the bottom of the board to the wall.
Use the Pythagorean Theorem.
10 ft. board
Simplify.
8 ft.wall
Possible solutions x
Check.
Both are solutions of the radical equation, but since the distance from the bottom of the board to the wall must be nonnegative, – 6 is not a solution of the problem.
The bottom of the board must be 6 feet from the wall.
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Example: Word Problem

10. Example: The time T (in seconds) taken for a pendulum of length L (in feet) to make one full swing, back and forth, is given by the formula
To the nearest hundredth, how long is a pendulum which takes 2 seconds to complete one full swing?
(3.24)
Radical equation
Check.
True.
Isolate the radical.
A pendulum of length approximately 3.24 feet will make one full swing in 2 seconds.
Square both sides.
Solve for L. (to the nearest hundredth)
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Example: Word Problem