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Answer to Question #22411 in Trigonometry for Kristen Woods

Question #22411
Solve the given equation.

11a. x^(2/3)+6x^(1/3)=7

11b. (y+3)^2-8(y+3)+12=0
Expert's answer
11a. x^(2/3)+6x^(1/3)=7
As we have here rational degrees, x must be non-negative:x>=0 Make a substitution
y = x^(1/3)> 0
Then
y^2 + 6y - 7 =0
D = 36 + 4*7 =64 = 8^2
y1 = (-6+8)/2 =1
y2 = (-6-8)/2 =-7 < 0

Since y>0, we obtain only one solution
y1 = 1 =x^(1/3)
whence
x=1

Answer: 1
---------------------------------

11b. (y+3)^2-8(y+3)+12=0

Make a substitution
t = y+3
Then
t^2 - 8t + 12 =0
D = 64 - 48 =16 = 4^2
t1 = (8+4)/2 =6
t2 = (8-4)/2 =2

Whence
y1 = t1-3 =6-3=3
y2 = t2-3 =2-3=-1

Answer: two roots: {-1, 3}

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