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

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

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

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

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

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

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

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

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

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

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

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

 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}

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

 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} 

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

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

Example  Solve SOLUTION CHECK 

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

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

Graph y = |x| Make T-table