1. Solving Rational Equations and Radical Equations
Section 3.4
Solving Rational Equations and Radical Equations

2. Objectives
Solve rational equations.
Solve radical equations.

3. Rational Equations
Equations containing rational expressions are called rational equations. Solving such equations requires multiplying both sides by the least common denominator (LCD) of all the rational expressions to clear the equation of fractions.

4. Example
Solve:
Multiply both sides by the LCD 6.

5. Example (continued)
The possible solution is 5. We check using a table in ASK mode. Since the value of is 0 when x = 5, the number 5 is the solution.

6. Example
Solve:
Multiply both sides by the LCD x  3.

7. Example (continued) The possible solutions are –3 and 3. Check x = –3:
TRUE
NOT DEFINED
The number 3 checks, so it is a solution.
Division by 0 is not defined, so 3 is not a solution.

8. Radical Equations
A radical equation is an equation in which variables appear in one or more radicands. For example:
The Principle of Powers
For any positive integer n:
If a = b is true, then an = bn is true.

9. Solving Radical Equations
To solve a radical equation we must first isolate the radical on one side of the equation. Then apply the Principle of Powers. When a radical equation has two radical terms on one side, we isolate one of them and then use the principle of powers. If, after doing so, a radical terms remains, we repeat these steps.

10. Example
Solve
Check x = 5:
TRUE
The solution is 5.

11. Example
Solve:
First, isolate the radical on one side.

12. Example (continued) The possible solutions are 9 and 2. Check x = 9.
TRUE
FALSE
Since 9 checks but 2 does not, the only solution is 9.

13. Example
Solve:

14. Example (continued)
We check the possible solution, 4, on a graphing calculator.
Since y1= y2 when x = 4, the number 4 checks. It is the solution.