# 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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