1. Start Up Day 38 12/15 Solve Each Equation:

2. OBJECTIVE: ESSENTIAL QUESTION: What is a Radical Equation? When you square each side of an equation, is the resulting equation equivalent to the original? S.W.B.A.T. Solve equations containing radicals. HOME LEARNING: P. 395 # 9, 12, 18, 22, 26, 29, 33, 38, 41, 49 & 52 MathXLforSchool 40 Point Mid-Chapter Quiz (Covering Lessons 6.1 through 6.4). Due by 1/6

3. Section 6.5 Radical Equations & Extraneous Solutions A radical equation is an equation in which the variable occurs in a square root, cube root, or any higher root. In this section, you will learn how to solve radical equations. When the variable occurs in a square root, it is necessary to square both sides of the equation. When you square both sides of an equation, sometimes extra answers creep in, called extraneous roots. For example, consider the following very simple original equation. Square both sides. Solve the equation. But -2 does not work in the original equation. Then - 2 is an extraneous root. It’s like a hitchhiker that we picked up when we squared both sides. It works only in the squared form and is not a root of the original.

4. Radical Equations Just to note then…. When you solve a rational equation and must raise both sides to an even power, remember to check your roots. Throw out any extra (extraneous) roots from your solution set. Another thing that you should know is … if your equation contains two or more square root expressions, you will need to isolate one square root expression, square both sides, and then you may have to repeat the process. You may have to square both sides twice.

5. Solving Radical Equations Solving Radical Equations Containing nth Roots 1) If necessary, arrange terms so that one radical is isolated on one side of the equation. 2) Raise both sides of the equation to the nth power to eliminate the nth root. 3) Solve the resulting equation. If this equation still contains radicals, repeat steps 1 and 2. 4) Check all proposed solutions in the original equation.

6. Solving Radical Equations EXAMPLE 1: Solve: SOLUTION 1) Isolate a radical on one side. Simplify. 2) Raise both sides to the nth power. Because n, the index, is 2, we square both sides.

7. Solving Radical Equations This is the equation from step 2. 3) Solve the resulting equation. Subtract 4 from both sides. Divide both sides by 3. 4) Check the proposed solution in the original equation. Check 20: true The solution is 20. The solution set is {20}. ? ? ?

8. Solving Radical Equations YOUR TURN: Solve: SOLUTION 1) Isolate a radical on one side. The radical can be isolated by adding 5 to both sides. We obtain Simplify. 2) Raise both sides to the nth power.

9. Solving Radical Equations 3) Solve the resulting equation. 4) Check the proposed solution in the original equation. Check 9: true The solution is 9. The solution set is {9}. ? ?

10. Solving Radical Equations EXAMPLE 2: Solve: SOLUTION 1) Isolate a radical on one side. The radical,, can be isolated by subtracting 11 from both sides. We obtain Simplify. 2) Raise both sides to the nth power. Because n, the index, is 2, we square both sides.

11. Solving Radical Equations This is the equation from step 2. 3) Solve the resulting equation. Subtract 5 from both sides. Divide both sides by 2. 4) Check the proposed solution in the original equation. Check 10: false Therefore there is no solution to the equation. ? ? ?

12. Solving Radical Equations EXAMPLE 3: Solve: SOLUTION 1) Isolate a radical on one side. The radical,, can be isolated by subtracting 3x from both sides. We obtain Simplify. Use the special formula 2) Raise both sides to the nth power. Because n, the index, is 2, we square both sides.

13. Solving Radical Equations Equation from step 2. 3) Solve the resulting equation. Because of the -term, the resulting equation is a quadratic equation. We need to write this quadratic equation in standard form. We can obtain zero on the left side by subtracting 3x and 7 from both sides. CONTINUED Subtract 3x and 7 from both sides. Factor out the GCF, 9. Divide both sides by 9. Factor the right side.

14. Solving Radical EquationsCONTINUED: Set each factor equal to 0. Solve for x. 4) Check the proposed solutions in the original equation. Check -2:Check -1: The solution is -1. The solution set is {-1}. truefalse ? ? ?? ? ?

15. Solving Radical Equations YOUR TURN: Solve: SOLUTION 4) Check the proposed solution in the original equation. Check 12: true ? ?

16. In Summary… Solving Radical Equations Containing nth Roots 1.Isolate one radical on one side of the equation. 2.Raise both sides to the nth power 3.Solve the resulting equation 4.Check proposed solutions in the original equation. Sometimes proposed solutions will work in the final simplified form of the original equation, but will not work in the original equation itself. These imposter roots that sometimes slip in when we square both sides of an equation are called extraneous roots.

17. Solving Radical Equations—Double Trouble! EXAMPLE 4: Solve: SOLUTION 1) Isolate a radical on one side. The radical,, can be isolated by subtracting from both sides. We obtain Simplify. Use the special formula 2) Raise both sides to the nth power. Because n, the index, is 2, we square both sides.

18. Solving Radical Equations Combine like terms. CONTINUED 1) Isolate a radical on one side. The radical, can be isolated by subtracting 20 + x from both sides and then dividing both sides by -8. We obtain 2) Raise both sides to the nth power. Because n, the index, is 2, we square both sides. Square the 3 and the

19. Solving Radical EquationsCONTINUED 3) Solve the resulting equation. This is the equation from the last step. Subtract 4 from both sides. 3) Check the proposed solution in the original equation. Check 5: The solution is 5. The solution set is {5}. true ? ? ?

20. Let’s try one more: p. 395 #18 Already Isolated…So Power UP! Why 3/2? Simplify Each Side. Remember: We Multiply Exponents to raise a power to a power! Simplify your radical expression. Solve your equation.