6-5 Solving Square Root and Other Radical Equations

Quick Review

To solve a radical equation, you must isolate a radical expression on one side of the equation. You can then rewrite the radical expression using a rational exponent and use the reciprocal of the exponent to solve the equation.

For example, to solve a square root equation, you square each side of the equation. Check all possible solutions in the original equation to eliminate extraneous solutions.

Example

What is the solution of 4 ( x − 2 ) 2 3 = 16 ? 4 . open , x minus 2 , close super 2 thirds end super . equals 16 question mark

( x − 2 ) 2 3 = 4 Isolate the radical . ( ( x − 2 ) 2 3 ) 3 2 = 4 3 2 Raise both sides to the  3 2  power . ( x − 2 ) 6 6 = 4 3 2 Law of exponents . | x − 2 | = 8 Simplify . table with 4 rows and 3 columns , row1 column 1 , open , x minus 2 , close super 2 thirds end super , column 2 equals 4 , column 3 cap isolatetheradical . . , row2 column 1 , open . open , x minus 2 , close super 2 thirds end super . close super 3 halves end super , column 2 equals , 4 super and 3 halves end super , column 3 cap raisebothsidestothe . 3 halves . power , . , row3 column 1 , open , x minus 2 , close super 6 sixths end super , column 2 equals , 4 super and 3 halves end super , column 3 cap lawofexponents . . , row4 column 1 , absolute value of , x minus 2 , end absolute value , , column 2 equals 8 , column 3 cap simplify , . , end table

x = 10 x equals 10  or x = − 6 x equals negative 6  Solve for x.

Exercises

Solve each equation. Check for extraneous solutions.

44. 2 + x + 5 = 4 2 plus , square root of x plus 5 end root , equals 4
45. 3 2 x + 6 = 18 3 . square root of 2 x plus 6 end root . equals 18
46. 5 ( 3 x + 1 ) 1 4 = 10 5 . open , 3 x plus 1 , close super 1 fourth end super . equals 10
47. 4 ( 3 x − 3 ) 2 3 = 36 4 . open , 3 x minus 3 , close super 2 thirds end super . equals 36
48. 3 x + 3 − 1 = x square root of 3 x plus 3 end root . minus 1 equals x
49. x + 6 + 2 = x + 6 square root of x plus 6 end root , plus 2 equals x plus 6
50. 5 x + 1 − 2 x = 1 square root of 5 x plus 1 end root . minus 2 square root of x equals 1
51. 2 x + 9 − x = 3 square root of 2 x plus 9 end root . minus square root of x equals 3
52. Electricity The power P, in watts, that a circular solar cell produces and the radius of the cell r in centimeters are related by the square root equation r = P 0.02 π . r equals . square root of fraction p , over 0.02 , pi end fraction end root . .  About how much power is produced by a cell with a radius of 12 cm?

6-6 Function Operations

Quick Review

When performing function operations, you can use the same rules you used for real numbers, but you must take into consideration the domain and range of each function. The composition of function g with function f is defined as ( g ∘ f ) ( x ) = g ( f ( x ) ) . open g composition f close open x close equals g open f open x close close .

Example

Let f(x) = x + 3 and g ( x ) = x 2 − 2 . g open x close equals , x squared , minus 2 .  What is ( g ∘ f ) ( − 2 ) ? open g composition f close open negative 2 close question mark

Therefore, ( g ∘ f ) ( − 2 ) = − 1 open g composition f close open negative 2 close equals negative 1

Exercises

Let f ( x ) = x − 4 f open x close equals x minus 4  and g ( x ) = x 2 − 16 . g open x close equals , x squared , minus 16 .  Perform each function operation and then find the domain.

53. f ( x ) + g ( x ) f open x close plus g open x close
54. g ( x ) − f ( x ) g open x close minus f open x close
55. f ( x ) · g ( x ) f open x close middle dot g open x close
56. g ( x ) f ( x ) fraction g , open x close , over f , open x close end fraction

Let g ( x ) = 5 x − 2 g open x close equals 5 x minus 2  and h ( x ) = x 2 + 1 . h open x close equals , x squared , plus 1 .  Find the value of each expression.

57. ( h ∘ g ) ( − 1 ) open h composition g close open negative 1 close
58. ( h ∘ g ) ( 0 ) open h composition g close open 0 close
59. ( g ∘ h ) ( 2 ) open g composition h close open 2 close
60. ( g ∘ h ) ( a ) open g composition h close open eh close
61. Discounts A grocery store is offering a 50% discount off a $4.00 box of cereal. You also have a $1.00 off coupon for the same cereal. Use a composite function to show whether it is better to use the coupon before or after the store discount.

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