1. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
§7.6 Radical Equations
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. 7.5 Review § Any QUESTIONS About Any QUESTIONS About HomeWork
MTH 55
Review §
Any QUESTIONS About
§7.5 → Rational Exponents
Any QUESTIONS About HomeWork
§7.5 → HW-28

3. Radical Equations
A Radical Equation is an equation in which at least one variable appears in a radicand.
Some Examples:

4. Power Rule vs Radical Eqns
Power Rule for Solving Radical Equations:
If BOTH SIDES of an equation are RAISED TO THE SAME POWER, ALL solutions of the original equation are ALSO solutions of the NEW equation

5. Caveat PowerRule → Check
CAUTION
Read the power rule carefully; it does not say that all solutions of the new equation are solutions of the original equation. They may or may not be…
Solutions that do not satisfy the original equation are called extraneous solutions; they must be discarded.  Thus the CHECK is CRITICAL

6. ReCall Exponent Power Rule
The Power Rule Provides a Crucial Tool for solving Radical Equations.
Recall the Exponent Power Rule
If a = b, then an = bn for any natural-number exponent n

7. Example  Solve by PwrRule
Solve Radical Equations:
a) b)
SOLUTION
a) b)
Check
Check
True
True

8. Example  Solve SOLUTION Check
4 Satisfies the original Eqn, so 4 is verified as a Solution

9. Solving Radical Equations
Isolate the radical.
If there is more than one radical term, then isolate one of the radical terms.
Raise both sides of the equation to the same power as the root index.
If all radicals have been eliminated, then solve. If a radical term remains, then isolate that radical term and raise both sides to the same power as its root index.
Check each solution. Any apparent solution that does not check is an extraneous solution

10. Example  Solve SOLUTION Square both sides. Use FOIL or Formula.
Subtract x from both sides.
Subtract 7 from both sides.
Factor.
Use the zero-products theorem.
The TENTATIVE Solutions

11. Example  Solve Check BOTH Tentative Solutions
False.
True.
Because 2 does not check, it is an extraneous solution. The only soln is 9

12. Example  Solve What Produced the Extraneous Solution?
At this step we Squared a NEGATIVE Number withOUT Knowing it…
Square both sides.
If x = 2, then (x−5) = −3
So Squaring (x−5) is the SAME as Squaring −3; we just didn’t know it
Thus 2 is a solution to
But NOT a solution to

13.  Example  Solve SOLUTION Check
This tentative solution x=4 does not check, so it is an extraneous solution. The equation has no solution; the solution set is {Ø}

14.  Example  Solve SOLUTION Check
So 13 checks. The solution set is {13}

15.  Example  Solve SOLN Check So 36 checks. The solution set is {36}
Isolate the variable radical
Using the Power Rule
Check
So 36 checks. The solution set is {36} 

16. Example  Solve SOLN Isolate the variable radical
Sq Both Sides to Remove Radical
(x−1)2 ≠ x2 −12
Apply Zero-Products
Tentative Solutions

17.   Example  Solve Check BOTH Tentative Solutions4 −1   4
3+1 −1
2+1
In this Case 4 checks while −1 does NOT. The solution set is {4}

18. Example  Solve
SOLUTION
CHECK 

19. WhiteBoard Work Problems From §7.6 Exercise Set
20, 26, 30, 46, 56
Remember, Raising Both Sides of Eqn to an EVEN Power can introduce EXTRANEOUS Solutions

20. All Done for Today
Life Expectancy

21. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
Appendix
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer –

22. Graph y = |x|
Make T-table