DOING BATTLE WITH RADICAL EQUATIONS EXAMPLES
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1 DOING BATTLE WITH RADICAL EQUATIONS EXAMPLES 1. Inverse Operations Operations that undo each other, such as multiplication and division.. Warm-up Activity Ask students to rewrite each of the following using inverse operations. a) x + 4 = 10 b) x 5 = 6 c) x = 1 x d) = 6 e) x + 4 = 10 x f) 5 = 1 a) x = 10 4 x = 10 4 b) x = x = c) x = 1 x = 1 x d) () = 6 () x = 6() e) x = 10 4 (x + 4 4) = (10 4) x = (10 4) x f) = x ( )() = (1 + 5)() x = (1 + 5)(). Ask students what operation represents the inverse (opposite) of taking square root. (Make sure that students understand that squaring and square roots are inverse operations. 4. Equations containing radicals with variables in the radicand are called radical equations. To solve such equations, first isolate the radical on one side of the equation. Then square each side of the equation to eliminate the radical. 5. Have students solve each equation below by squaring each side of the equation. (Make sure that students understand that we are using an inverse operation.) a) x = 4 ( x ) = (4) x = 16 b) m = ( m ) = () m = 9 c) x = 16 ( x ) = (16) x = 56 c) x = ( x ) = ( ) 4 x = 16 1 Doing Battle with Radical Equations 00 Rev
2 6. Emphasize the importance of checking solution! x = - 4 ( x ) = (-4) x = 16 x = = Does not check. 7. Example: Solve x = 11 and check to see if answer is reasonable! x = 11 ( x ) = (11) x = 11 Check: x = = = Example: Solve x = -8 and check to see if answer is reasonable! x = -8 ( x ) = (-8) x = 64 x = Check: x = -8 () = = Example: Solve a + 4 = and check to see if answer is reasonable! a + 4 = ( a + 4 ) = () a + 4 = 4 x = 0 Check: a + 4 = = 4= = Doing Battle with Radical Equations 00 Rev
3 x 10. Example: Solve = -5 and check to see if answer is reasonable! ( x = -5 x ) = (-5) x = 5 x = 75 x = 7.5 x Check: = -5 (7.5) = -5 5 = Example: Solve x 5-4 = 0 and check to see if answer is reasonable! x 5-4 = 0 x 5 = 4 ( x 5 ) = (4) x 5 = 16 x = 1 x = 7 Check: x 5-4 = 0 (7) 5-4 = = = 0 0 = 0 1. Why must you check solutions? Consider this equation. x = Now square each side. x = 4 The solutions for this equation are and. We know that x =, therefore, one of the solutions is not correct. We must check the answer to see if both are reasonable! Doing Battle with Radical Equations 00 Rev
4 1. Stress that squaring each side of an equation often produces results that do not check. Thus, it is important to check all possible solutions. A result that does not check is called an extraneous value. 14. Emphasize the importance of isolating the radical by showing an example of what happens if you do not. x + = 0 ( x + ) = (0) x + 6 x + 9 = 0 The radical still remains! 15. Extension The time t in seconds that it takes an object, initially at rest, to fall a vertical distance of s v meters is given by the formula ( g is the acceleration due to gravity.) Find s v, when t = 4 and the acceleration due to gravity is 9.8 m. (answer = 78.4 meters). s Doing Battle with Radical Equations 00 Rev
5 Name: Date: Class: DOING BATTLE WITH RADICAL EQUATIONS WORKSHEET Solve by using inverse operations. Check! 1. x =. x = -. x = 5 4. x = 5. x = x - 5 = 0 7. x + 1 = x = x + 1 = x = x = x - = x 8 = 0 Doing Battle with Radical Equations 00 Rev
6 DOING BATTLE WITH RADICAL EQUATIONS WORKSHEET KEY Solve by using inverse operations. Check! 1. x =. x = - ( x ) = () x = 9 Check x = 9 = =. x = 5 ( x ) = ( x = 4 ) Check x = - 4 = - - ( x ) = (5) x = 65 Check x = 5 65 = 5 5 = 5 4. x = ( x ) = x = 9 x = () Check x = () = = 5. x = -9 ( x ) = ( x = 81 x = 7 9) Check x = -9 (7) = x - 5 = 0 ( x ) = x = 5 (5) Check x - 5 = = 0 0 = 0 Doing Battle with Radical Equations 00 Rev
7 7. x + 1 = 0 x = -1 ( x ) = (-1) x = 1 x =.5 Check (.5) = x = x = 8 x = ( x ) = () x = 9 x = 4.5 Check 5 + (4.5) = = 8 8 = x + 1 = ( 4 x + 1 ) = () 4x + 1 = 9 4x = 8 x = Check 4 () + 1 = 9 = = x = 0 8 x +1 = 5 ( 8 x + 1 ) = (5) 8x + 1 = 5 8x = 4 x = Check 8 () + 1 = 5 5 = 5 5 = 5 Doing Battle with Radical Equations 00 Rev
8 11. x = 6 4 ( x ) = (6) 4 x = 6 4 x = 144 Check 144 = = 6 6 = x - = 0 ( 4x = 4x ) = () 4x = 4 4x = 1 x = Check 4() 4 - = 0 = 0 0 = 0 - = x 8 = 0 x 8 = 4 x 8) = 4 ( x 8 = 16 x = 44 x = x = or Check 5 ( ) 8 = = 0 16 = 4 4 = 4 Check 5 ( ) 8 = = 0 16 = 4 4 = 4 Doing Battle with Radical Equations 00 Rev
9 Student Name: Date: DOING BATTLE WITH RADICAL EQUATIONS CHECKLIST 1. On question 1, did the student solve and check the equation correctly by using a. Yes (5 points) (0 points). On question, did the student solve and check the equation correctly by using a. Yes (5 points) (0 points). On question, did the student solve and check the equation correctly by using a. Yes (5 points) (0 points) Doing Battle with Radical Equations 00 Rev
10 4. On question 4, did the student solve and check the equation correctly by using a. Yes (5 points) (0 points) 5. On question 5, did the student solve and check the equation correctly by using a. Yes (5 points) (0 points) 6. On question 6, did the student solve and check the equation correctly by using a. Yes (5 points) (0 points) Doing Battle with Radical Equations 00 Rev
11 7. On question 7, did the student solve and check the equation correctly by using a. Yes (5 points) (0 points) 8. On question 8, did the student solve and check the equation correctly by using a. Yes (5 points) (0 points) 9. On question 9, did the student solve and check the equation correctly by using a. Yes (5 points) (0 points) Doing Battle with Radical Equations 00 Rev
12 10. On question 10, did the student solve and check the equation correctly by using a. Yes (5 points) (0 points) 11. On question 11, did the student solve and check the equation correctly by using a. Yes (5 points) (0 points) 1. On question 1, did the student solve and check the equation correctly by using a. Yes (5 points) (0 points) Doing Battle with Radical Equations 00 Rev
13 1. On question 1, did the student solve and check the equation correctly by using a. Yes (5 points) (0 points) Doing Battle with Radical Equations 00 Rev
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