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Hawkes Learning Systems: College AlgebraSection 2.6: Radical Equations




Objectives

o Solving radical equations.o Solving equations with positive rational exponents.




Radical Equations

o A radical equation is an equation that has at least one radical expression containing a variable, while any non-radical expressions are polynomial terms.

o One reasonable approach to solving these equations is to raise both sides of the equation to whatever power is necessary to “undo” the radical(s).

o This may result in some extraneous solutions that we must identify and discard by checking all of our eventual solutions in the original equation.




Radical Functions

For example, consider the equation

From this, we obtain the solution that . We have gained a second (and false) solution.

4x 2 16x Square both sides.

16x Square root both sides.

or 4 4 x x




Solving Radical Equations

Method of Solving Radical Equations

Step 1: Begin by isolating the radical expression on one side of the equation. If there is more than

one radical expression, choose one of the radical expressions to isolate on one side.

Step 2: Raise both sides of the equation by the power necessary to “undo” the isolated radical. That

is, if the radical is an root, raise both sides to the power.

thnthn





Method of Solving Radical Equations (Cont.)

Step 3: If any radical expressions remain, simplify the equation if possible and then repeat steps 1

and 2 until the result is a polynomial equation. When a polynomial equation has been obtained, solve the equation using

polynomial methods.

Step 4: Check your solutions in the original equation. Any extraneous solutions must be discarded.




Example 1: Solving Radical Equations

Solve the radical equation. 2 2x x

2 2x x

2 2

2 2x x 22 4 4x x x 20 3 2x x

0 1 2x x

1, 2x

Step 1: Isolate the radical expression.

Step 2: Square both sides.

Step 3: Solve the polynomial equation.

Note: Both roots satisfy the original equation.





Solve the radical equation.3 4 1x x

2 2

3 1 4x x

3 1 2 4 4x x x

2 4 2x

2 2

4 1x

4 1x 3x





Solve the radical equation.24 1 1 0x x

4

424 1 1x x 2 1 1x x

2 2 0x x

2 1 0x x

2, 1x





Solve the radical equation.

1 1x x 1 1x x

2 2

1 1x x 21 2 1x x x 20 3x x 0 3x x

0, 3x 0x

Note that , so -3 is an extraneous solution.

1 ( 3) ( 3) 1





Caution!Substitute solutions back into the original equation to check them! Some solutions may not fit, and are therefore extraneous.




Solving Equations with Positive Rational Exponents

o Equations containing terms with positive rational exponents can be viewed as radical equations.

o Rewriting each term that has a positive rational exponent as a radical will allow us to use the method developed previously to solve rational equations.




Example 4: Equations with Positive Rational Exponents

Solve the following equation with a rational exponent.2

3 16 0x 2

3 16x

2 316x 31

216x

34x

64x

Step 1: Isolate the term containing the rational exponent.

Step 2: Cube both sides to eliminate the cubed root.

Step 3: Take the square root of both sides.

Step 4: Verify that both numbers satisfy the original equation.

3 2 16x




Example 5: Equations with Positive Rational Exponents

Solve the following equation with a rational exponent.1

2 4(7 21 130) 4x x 24 7 21 130 4x x 2 47 21 130 4x x 27 21 130 256x x 27 21 126 0x x

27 3 18 0x x 2 3 18 0x x

3 6 0x x 3,6x

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