0=3x^2+3x-1

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Solution for 0=3x^2+3x-1 equation:


Simplifying
0 = 3x2 + 3x + -1

Reorder the terms:
0 = -1 + 3x + 3x2

Solving
0 = -1 + 3x + 3x2

Solving for variable 'x'.

Combine like terms: 0 + 1 = 1
1 + -3x + -3x2 = -1 + 3x + 3x2 + 1 + -3x + -3x2

Reorder the terms:
1 + -3x + -3x2 = -1 + 1 + 3x + -3x + 3x2 + -3x2

Combine like terms: -1 + 1 = 0
1 + -3x + -3x2 = 0 + 3x + -3x + 3x2 + -3x2
1 + -3x + -3x2 = 3x + -3x + 3x2 + -3x2

Combine like terms: 3x + -3x = 0
1 + -3x + -3x2 = 0 + 3x2 + -3x2
1 + -3x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
1 + -3x + -3x2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-0.3333333333 + x + x2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + x + 0.3333333333 + x2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + x + x2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + x + x2 = 0 + 0.3333333333
x + x2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
x + x2 = 0.3333333333

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + x2 = 0.3333333333 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 0.3333333333 + 0.25

Combine like terms: 0.3333333333 + 0.25 = 0.5833333333
1.25 + x2 = 0.5833333333

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.5833333333

Calculate the square root of the right side: 0.763762616

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.763762616 and -0.763762616.

Subproblem 1

x + 0.5 = 0.763762616 Simplifying x + 0.5 = 0.763762616 Reorder the terms: 0.5 + x = 0.763762616 Solving 0.5 + x = 0.763762616 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.763762616 + -0.5 x = 0.763762616 + -0.5 Combine like terms: 0.763762616 + -0.5 = 0.263762616 x = 0.263762616 Simplifying x = 0.263762616

Subproblem 2

x + 0.5 = -0.763762616 Simplifying x + 0.5 = -0.763762616 Reorder the terms: 0.5 + x = -0.763762616 Solving 0.5 + x = -0.763762616 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.763762616 + -0.5 x = -0.763762616 + -0.5 Combine like terms: -0.763762616 + -0.5 = -1.263762616 x = -1.263762616 Simplifying x = -1.263762616

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.263762616, -1.263762616}

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