if ×= 2+ 2⅔+2⅓ ,then x³-6x²+6x
Answers
Answer:
x³ - 6x² + 6x = 2
Step-by-step explanation:
Given---> x = 2 + 2²/³ + 2¹/³
To find---> Value of x³ - 6x² + 6x
Solution---> We know that,
( a + b )³ = a³ + b³ + 3ab ( a + b ) , and
(a - b )³ = a³ - b³ - 3ab ( a - b )
Now, x = 2 + 2²/³ + 2¹/³
=> x - 2 = 2²/³ + 2¹/³
Cubing both sides , we get
=> ( x - 2 )³ = ( 2²/³ + 2¹/³ )³
Now applying above identity , we get
=> x³ - 8 - 3 (x) ( 2 ) (x - 2 ) = (2²/³)³ + (2¹/³)³
+ 3 × 2¹/³ 2²/³ (2¹/³ + 2²/³ ) We have a law of exponent
( aᵐ )ⁿ = aᵐⁿ , applying it here , we get
=> x³ - 8 -6x(x - 2 ) = 2² + 2¹ + 3 × 2¹ ( x - 2 )
=> x³ - 8 - 6x² + 12x = 4 + 2 + 6x - 12
=> x³ - 6x² + 12x - 8 = 6x - 6
=> x³ - 6x² + 12x - 6x -8 + 6 = 0
=> x³ - 6x² + 6x - 2 = 0
=> x³ - 6x² + 6x = 2
Answer:
Step-by-step explanation:
=>x=2+2⅔+2⅓
=>x-2=2⅔+2⅓. ......(equ 1)
cubing both sides
=>(x-2)³=(2⅔+2⅓)³
=>x³-8-3.2x(x-2) = (2⅔)³+(2⅓)³+3.2⅔.2⅓(2⅔+2⅓)
=>x³-8-6x(x-2) = (2)²+(2)¹+3.2^(⅔+⅓)(2⅔+2⅓)
=>x³-8-6x(x-2) = 4+2+3.2¹(2⅔+2⅓)
=>x³-8-6x²+12=6+6(x-2). ..(from equ 1.)
=>x³-6x²+4=6+6x-12
=>x³-6x²-6x-2=0
(... proved...)