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What is the solution of 3x^3 - 1 = 0

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jimb840 | Student, Grade 9 | (Level 1) Salutatorian

Posted September 4, 2013 at 2:40 PM via web

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What is the solution of 3x^3 - 1 = 0

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted September 4, 2013 at 4:21 PM (Answer #2)

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The equation 3x^3 - 1 = 0 has to be solved for x.

3x^3 - 1 = 0

=> 3x^3 = 1

=> `x^3 = 1/3`

=> `x = root(3) (1/3)`

In addition, the equation has two complex roots.

`((root(3)3)*x)^3 - 1 = 0`

=> `((root(3)3)*x - 1)((root(3)3)^2*x^2 + root(3)3x + 1) = 0`

`(root(3)3)*x - 1 = 0`

=> `x = 1/root(3) 3`

`((root(3)3)^2*x^2 + root(3)3x + 1) = 0`

=> x = `(-3^(1/3) +- sqrt(3^(2/3) - 4*3^(2/3)))/(2*3^(2/3))`

=> x = `(-1 +- sqrt(1 - 4))/(2*3^(1/3))`

=> x = `(-1 +- sqrt3*i)/(2*3^(1/3))`

The roots of the equation `3x^3 - 1 = 0` are `x = 3^(1/3)` , `x = (-1 - sqrt3*i)/(2*3^(1/3))`  and `x =(-1 + sqrt3*i)/(2*3^(1/3))`

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baxthum8 | High School Teacher | (Level 3) Associate Educator

Posted September 4, 2013 at 2:44 PM (Answer #1)

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Solution to `3x^3 - 1 = 0`

Add 1 to both sides.

`3x^3 = 1`

Divide both sides by 3

`x^3 = 1/3`

Take cubic root of each side.

`x =root(3)(1/3)`

Simplify:

`x = (root(3)(3))/3`

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