3x^2-1=6x-3

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Solution for 3x^2-1=6x-3 equation:


Simplifying
3x2 + -1 = 6x + -3

Reorder the terms:
-1 + 3x2 = 6x + -3

Reorder the terms:
-1 + 3x2 = -3 + 6x

Solving
-1 + 3x2 = -3 + 6x

Solving for variable 'x'.

Reorder the terms:
-1 + 3 + -6x + 3x2 = -3 + 6x + 3 + -6x

Combine like terms: -1 + 3 = 2
2 + -6x + 3x2 = -3 + 6x + 3 + -6x

Reorder the terms:
2 + -6x + 3x2 = -3 + 3 + 6x + -6x

Combine like terms: -3 + 3 = 0
2 + -6x + 3x2 = 0 + 6x + -6x
2 + -6x + 3x2 = 6x + -6x

Combine like terms: 6x + -6x = 0
2 + -6x + 3x2 = 0

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

Divide each side by '3'.
0.6666666667 + -2x + x2 = 0

Move the constant term to the right:

Add '-0.6666666667' to each side of the equation.
0.6666666667 + -2x + -0.6666666667 + x2 = 0 + -0.6666666667

Reorder the terms:
0.6666666667 + -0.6666666667 + -2x + x2 = 0 + -0.6666666667

Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000
0.0000000000 + -2x + x2 = 0 + -0.6666666667
-2x + x2 = 0 + -0.6666666667

Combine like terms: 0 + -0.6666666667 = -0.6666666667
-2x + x2 = -0.6666666667

The x term is -2x.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x + 1 + x2 = -0.6666666667 + 1

Reorder the terms:
1 + -2x + x2 = -0.6666666667 + 1

Combine like terms: -0.6666666667 + 1 = 0.3333333333
1 + -2x + x2 = 0.3333333333

Factor a perfect square on the left side:
(x + -1)(x + -1) = 0.3333333333

Calculate the square root of the right side: 0.577350269

Break this problem into two subproblems by setting 
(x + -1) equal to 0.577350269 and -0.577350269.

Subproblem 1

x + -1 = 0.577350269 Simplifying x + -1 = 0.577350269 Reorder the terms: -1 + x = 0.577350269 Solving -1 + x = 0.577350269 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 0.577350269 + 1 Combine like terms: -1 + 1 = 0 0 + x = 0.577350269 + 1 x = 0.577350269 + 1 Combine like terms: 0.577350269 + 1 = 1.577350269 x = 1.577350269 Simplifying x = 1.577350269

Subproblem 2

x + -1 = -0.577350269 Simplifying x + -1 = -0.577350269 Reorder the terms: -1 + x = -0.577350269 Solving -1 + x = -0.577350269 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -0.577350269 + 1 Combine like terms: -1 + 1 = 0 0 + x = -0.577350269 + 1 x = -0.577350269 + 1 Combine like terms: -0.577350269 + 1 = 0.422649731 x = 0.422649731 Simplifying x = 0.422649731

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.577350269, 0.422649731}

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