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1. 1. 7.7 Solving Radical Equations p.453
2. 2. What is a Radical Expression? • A Radical Expression is an equation that has a variable in a radicand or has a variable with a rational exponent. 3 x 10 2 3 yes 25 yes 3 x 10 no ( x 2)
3. 3. Steps to solve a radical equation: STEP 1: Isolate the radical on one side of the equation (if possible) STEP 2: Raise each side of the equation to a power equal to the index of the radical to eliminate the radical STEP 3: Solve the remaining polynomial equation. CHECK YOUR RESULTS.
4. 4. EXAMPLE – Solving a Radical Equation 5x 1 6 0 5x 1 6 2 2 5x 1 (6 ) 5x 1 36 5x 35 x 7 Square both sides to get rid of the square root
5. 5. EXAMPLE x 15 3 x 2 2 x 15 (3 x 15 (3 x 15 24 3 16 x) x )(3 9 6 x 15 16 15 x) x NO SOLUTION Since 16 doesn’t plug in as a solution. 9 6 x 6 x 4 x 16 x 1 3 4 1 1 Let’s Double Check that this works Note: You will get Extraneous Solutions from time to time – always do a quick check
6. 6. Let’s Try Some 2 3x 2 6 2( x 2) 2 3 50
7. 7. Let’s Try Some 2 3x 2 6 2( x 2) 2 3 50
8. 8. SOLVING MORE COMPLEX EQUATIONS 4( x 1) 2 101 20 2 4( x 1) 2 ( x 1) ( x 1) + because we are taking an even power (square root) of both sides 81 81 ( x 1) i ( x 1) ( x 1) 9i x 1 9i 2 2 4 81 4 81 1 4 i 92 9i 2 1 9i 2
9. 9. RADICAL EQUATIONS 3(5n 1) 1 3 2 3(5n 1) 1 3 2 1 3 (5n 1) 1 3 (5n 1) 3 (5n 1) 5n 5n 8 0 2 8 27 5n RAISE BOTH SIDES TO RECIPROCAL POWER 3 2 27 8 ISOLATE RADICAL / RATIONAL 3 3 SOLVE FOR THE VARIABLE 27 1 27 35 27 27 n 7 27
10. 10. SOLVING MORE COMPLEX EQUATIONS (2 x 1) 0.5 (3x 4) 0.25 0 0.5 0.25 (2 x 1) (3x 4) 0.5 4 0.25 4 [( 2 x 1) ] [(3x 4) ] 2 (2 x 1) 3 x 4 4x 2 4 x 1 3x 4 4x 2 Raise each side to the 4th power. This will get you integer powers – much easier to work with! x x 3 0 3 , 1 4 Factor
11. 11. Let’s Try Some . . . check for extraneous solutions (2 x 1) 0.5 (3x 4) 0.25 0
12. 12. Let’s Try Some . . . check for extraneous solutions (2 x 1) 0.5 (3x 4) 0.25 0 x 3 , 1 4
13. 13. Can graphing calculators help? SURE! x 1. 2. 3. 4. x 2 Input x for Y1 Input x-2 for Y2 Graph Find the points of intersection One Solution at (4, 2) To see if this is extraneous or not, plug the x value back into the equation. Does it work?
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