if x=2+2^1/3+4^1/3 show that x^3-6x^2+6x-2=0
Answers
Step-by-step explanation:
Given that x = 2 + 2^(2/3) + 2^(1/3)
so x - 2 = 2^(2/3) + 2^(1/3)
=(x-2)^3 = [2^(2/3)+2^(1/3)]^3
use (a-b)^3= a^3 - b^3 - 3ab(a-b)
and (a+b)^3= a^3 + b^3 + 3ab(a+b) formulae.
or x^3 - 8 - 6x(x-2) = 2^2 + 2^1 + 3*[2^{(2/3)+(1/3)}[2^(2/3)+2^(1/3)
or x^3 - 8 - 6x^2 + 12x = 4 + 2 + 6(x-2)
or x^3 - 8 -6x^2 + 12x = 6 + 6x - 12
or x^3 - 6x^2 +6x = 2
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Shambhu Bhat
Answered 2 years ago
What is the value of x^3 - 6x^2 +6x if x = 2 + 2^1/3 + 2^2/3?
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What is the value of x3−6x2+6x if x=2+21/3+22/3 ?
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Let 213=t
x=2+t+t2⟹x−2=t+t2
(x−2)3=(t+t2)3=t6+3t5+3t4+t3=4+6t2+6t+2=6+6(x−2)Replace t3 by 2
(x−2)3−6(x−2)=6 now expand
x3–6x2+12x−8–6x+12=6
x3–6x2+6x=2
Kanishk Saini
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Subhasish Debroy
Answered 3 years agoShivaram KratosPiyush Karmakar
From the given equation , we get
x- 2 = 2^2/3 + 2^1/3 , taking cube on both side,
( x -2)^3 = (2^2/3 +2^1/3)^3
Or, x^3 - 3*x^2 *2 +3*x* 2^2 -2^3 =( 2^2/3)^3 + (2^1/3)^3 + 3*2^2/3 * 2^1/3 (2^2/3 + 2^1/3)
or, x^3 -6x^2 +12x -8 = 4 +2+3*2^(2/3+1/3) * (x-2)
or, x^3 - 6x^2 +12x =8 +6+6(x-2)
or, x^3 -6x^2 +12x -6x = 14 -12
or, x^3 -6x^2 +6x =2
Therefore the value of x^3 -6x^2 +6x is 2
Ved Prakash Sharma
Answered 3 years agoShivaram Kratos
x=2+2^2/3+2^1/3 , Let 2^1/3=p
x=2+p^2+p
x-2=p^2+p……………….(1)
cubing both sides
x^3–8–3.x.2.(x-2)=p^6+p^3+3.p^2.p.(p^2+p) , put (p^2+p)= (x-2)from eq.(1).
x^3–8–6x^2+12x=p^6+p^3+3p^3.(x-2)
Put p^3=2 ,p^6=4
x^3–8–6x^2+12x=4+2+3×2×(x-2)
x^3–6x^2+12x=8+6+6x-12
x^3–6x^2+12x-6x=14–12
x^3–6x^2+6x=2 , answer.
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Anikait Panigrahi
Answered 2 years ago
From the given equation , we get
x- 2 = 2^2/3 + 2^1/3 , taking cube on both side,
( x -2)^3 = (2^2/3 +2^1/3)^3
Or, x^3 - 3*x^2 *2 +3*x* 2^2 -2^3 =( 2^2/3)^3 + (2^1/3)^3 + 3*2^2/3 * 2^1/3 (2^2/3 + 2^1/3)
or, x^3 -6x^2 +12x -8 = 4 +2+3*2^(2/3+1/3) * (x-2)
or, x^3 - 6x^2 +12x =8 +6+6(x-2)
or, x^3 -6x^2 +12x -6x = 14 -12
or, x^3 -6x^2 +6x =2
Therefore the value of x^3 -6x^2+6 x is 2
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Abhinav Taya
Answered 2 years ago
From the given equation , we get
x- 2 = 2^2/3 + 2^1/3 , taking cube on both side,
( x -2)^3 = (2^2/3 +2^1/3)^3
Or, x^3 - 3*x^2 *2 +3*x* 2^2 -2^3 =( 2^2/3)^3 + (2^1/3)^3 + 3*2^2/3 * 2^1/3 (2^2/3 + 2^1/3)
or, x^3 -6x^2 +12x -8 = 4 +2+3*2^(2/3+1/3) * (x-2)
or, x^3 - 6x^2 +12x =8 +6+6(x-2)
or, x^3 -6x^2 +12x -6x = 14 -12
or, x^3 -6x^2 +6x =2
Therefore the value of x^3 -6x^2 +6x is 2
Arshmeen Kaur
Answered 2 years ago
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Ankita Kannojia (Ankitanju)
Updated 2 years ago
x=2+2*2/3+2*1/3
x=2+(4/3+2/3)=(6+4+2)/3=12/3=4
x=4.
now put the value in the given expression to get the required value.
x^3-6x^2+6x=(4)^3–6(4)^2+6*4
=64–6*16+24=64–96+24
=-8.
Hence the value is -8.
Suraj Sharma
Answered 3 years ago
Here , given
x = 2+2^2/3+2^1/3
x = 2+(4/3+2/3)
x = 2+(4+2)/3
x = 2+6/3
x = 2+2
x = 4
now put the value in the given expression to get the required value
x^3–6x^2+6x
=(4)^3–6(4)^2+6*4
=64–6*16+24
=64–96+24
=88–96
= -8
Hence the value is -8.
Shraddha Parab
Answered 2 years ago
From the given equation , we get
x- 2 = 2^2/3 + 2^1/3 , taking cube on both side,
( x -2)^3 = (2^2/3 +2^1/3)^3
Or, x^3 - 3*x^2 *2 +3*x* 2^2 -2^3 =( 2^2/3)^3 + (2^1/3)^3 + 3*2^2/3 * 2^1/3 (2^2/3 + 2^1/3)
or, x^3 -6x^2 +12x -8 = 4 +2+3*2^(2/3+1/3) * (x-2)
or, x^3 - 6x^2 +12x =8 +6+6(x-2)
or, x^3 -6x^2 +12x -6x = 14 -12
or, x^3 -6x^2 +6x =2
Therefore the value of x^3 -6x^2 +6x is 2
Puneeth Kumar R
Answered 2 years ago
x=2+2*2/3+2*1/3
x=2+2*2*3/3+2*1
x=4*2*3/5*1
Therefore,
x=24/5=4.8
————————————————————————————————————
The value of x^3-6x^2+6x
=4.8^3–6*4.8^2+6*4.8
=110.592–138.24+28.8
=1.152