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Equations and Inequalities

Definition

Radical Equations
are equations in which the variable is inside a radical.

Examples:

A radical equation may be transformed into a simple linear or quadratic
equations. Sometimes the transformation process yields extraneous
solutions
. These are apparent solutions that may solve the transformed
problem but are not solutions of the original radical equation.

Example (1)

Solve the equation

Transform the radical equation into a linear equation ...

The solution set is {7}.

Example (2)

Solve the equation

Transform the radical equation into a quadratic equation ...

The solution set is {3}.

Example (3)

Solve the equation

The solution set is {2}.

PROCEDURE FOR SOLVING RADICAL EQUATIONS

Step 1: Isolate the term with a radical on one side.
Step 2: Raise both (entire) sides of the equation to the power that
will eliminate this radical , and simplify the equation.
Step 3: If a radical remains, repeat steps 1 and 2.
Step 4: Solve the resulting linear or quadratic equation.
Step 5: Check the solutions and eliminate any extraneous solutions.

Equations that are higher order or that have fractional powers often can
be transformed into a quadratic equation by introducing a u- substitution .
We say that equations are quadratic in form.

ORIGINAL EQUATION SUBSTITUTION NEW EQUATION

PROCEDURE FOR SOLVING EQUATIONS QUADRATIC IN
FORM


Step 1: Identify the substitution.
Step 2: Transform the equation into a quadratic form.
Step 3: Solve the quadratic equation.
Step 4: Apply the substitution to rewrite the solution in terms of
the original variable .
Step 5: Solve the resulting equation.
Step 6: Check the solutions in the original solutions.

Example (4)

Find the solution to the equation

The solution set is .

Example (5)

Find the solution to the equation .

The solution set is {-8, 125}.

Some equations (both polynomial and with rational exponents ) that are
factorable can be solved using the zero product property.

Example (6)

Solve the equation

The solution set is {-1, 0,4}.

Example (7)

Solve the equation

The solution set is {-2, -1, 1}.

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