1. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §7.6 Radical Equations

2. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §7.5 → Rational Exponents  Any QUESTIONS About HomeWork §7.5 → HW-34 7.5 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 3 Bruce Mayer, PE Chabot College Mathematics Radical Equations  A Radical Equation is an equation in which at least one variable appears in a radicand.  Some Examples:

4. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 4 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 5 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 6 Bruce Mayer, PE Chabot College Mathematics Power Rule  The Power Rule Provides a Crucial Tool for solving Radical Equations.  Recall the Power Rule If a = b, then a n = b n for any natural-number exponent n

7. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  Solve by PwrRule  Solve Radical Equations: a)b)  SOLUTION a)b) Check True Check True

8. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLUTION  Check  4 Satisfies the original Eqn, so 4 is verified as a Solution

9. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 9 Bruce Mayer, PE Chabot College Mathematics Solving Radical Equations 1.Isolate the radical.  If there is more than one radical term, then isolate one of the radical terms. 2.Raise both sides of the equation to the same power as the root index. 3.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. 4.Check each solution. Any apparent solution that does not check is an extraneous solution

10. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLUTION Square both sides. Use FOIL or Formula. Subtract x from both sides. Factor. Use the zero-products theorem. Subtract 7 from both sides. The TENTATIVE Solutions

11. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  Solve  Check BOTH Tentative Solutions False. True.  Because 2 does not check, it is an extraneous solution. The only soln is 9

12. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 12 Bruce Mayer, PE Chabot College Mathematics 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} 

13. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLUTION  Check  So 13 checks. The solution set is {13} 

14. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLN  Check  So 9 checks. The solution set is {9}  Using the Power Rule Isolate the variable radical

15. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 15 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLN Isolate the variable radical Sq Both Sides to Remove Radical (x−1) 2 ≠ x 2 −1 2 Apply Zero-Products Tentative Solutions

16. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 16 Bruce Mayer, PE Chabot College Mathematics Example  Solve   Check BOTH Tentative Solutions 3+1 4 4 2+1 −1−1 −1−1   In this Case 4 checks while −1 does NOT. The solution set is {4}

17. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 17 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLUTION  CHECK 

18. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 18 Bruce Mayer, PE Chabot College Mathematics 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

19. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 19 Bruce Mayer, PE Chabot College Mathematics All Done for Today Life Expectancy

20. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 20 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

21. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 21 Bruce Mayer, PE Chabot College Mathematics Graph y = |x|  Make T-table

22. BMayer@ChabotCollege.edu MTH55_Lec-45_7-6a_Radical_Equations.ppt 22 Bruce Mayer, PE Chabot College Mathematics