Section 6.6: Solve Radical Equations Starter: CC 6.5 Part 2

 Before beginning, it may be easier to rewrite as rational exponents.  Step 1) Isolate the radical.  Get the radical by itself  Step 2) Raise both sides to the reciprocal power  √x use (√x) 2, 3 √x use ( 3 √x) 3  Step 3) Simplify, use foil if necessary  Step 4) Solve for the variable.  Step 5) Check all solutions, some may be extraneous. To Solve Radical Equations:

 √(5x + 1) = 6  (5x + 1) 1/2 = 6Rewrite as rational power  [(5x + 1) 1/2 ] 2 = 6 2 Square both sides  5x + 1 = 36Simplify  5x = 35Subtract 1  x = 7Divide by 5  (5(7) + 1) 1/2 = 6Substitute and check.  (36) 1/2 = 6  6=6 Ex. √(5x + 1) = 6

 3 √x – 10 = - 3  (x) 1/3 – 10 = - 3Rewrite  (x) 1/3 = Add 10  (x) 1/3 = 7Simplify  [(x) 1/3 ] 3 = 7 3 Cube both sides  x = 343Simplify  3 √x – 10 = - 3Substitute and check  3 √343 – 10 = - 3  = -3  - 3 = - 3 EX. 3 √x – 10 = -3

 3x 3/2 + 5 = 380  3x 3/2 = 380 – 5Subtract 5  3x 3/2 = 375Simplify  x 3/2 = 375/3 = 125Divide by 3 and simplify  [x 3/2 ] 2/3 = 125 2/3 Raise both sides the reciprocal power  x = 5 2 = 25Simplify. EX. 3x 3/2 + 5 = 380

 Step 1) Isolate one radical  Step 2) Raise both sides to the reciprocal power  Step 3) Simplify  Step 4) Isolate the other radical  Step 5) Raise to the reciprocal power  Step 6) Solve by simplifying. When there are two radicals…

 √(x + 6) – 2 = √(x - 2)  [√(x + 6) – 2] 2 = [√(x - 2) ] 2 Square both sides  (x + 6) – 4 √(x + 6) + 4 = x – 2Foil the left, simplify the right  x + 10 – 4 √(x + 6) = x – 2 Simplify and isolate the radical  -4√(x + 6) = x – 2 – (x + 10)Subtract (x + 10)  √(x + 6) = -12/-4 = 3Divide by (-4)  √(x + 6) = 3Simplify  [√(x + 6)] 2 = (3) 2 Square both sides  x + 6 = 9Simplify  x = = 3Subtract 6 EX. √(x + 6) – 2 = √(x - 2)

 √(3 + 6) – 2 = √(3 - 2)Substitute and check  √9 – 2 = √1  3-2 = 1  1 = 1 Always Check your answers!

 456:  A: 3, 7, 9, 13, 15, 17, 23, 25  B: 11, 19, 21, 25, 27, 37  C: 18, 26, 28, 38, 42 Your Turn: