3x^2=6x+1

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


Simplifying
3x2 = 6x + 1

Reorder the terms:
3x2 = 1 + 6x

Solving
3x2 = 1 + 6x

Solving for variable 'x'.

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

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

Combine like terms: 1 + -1 = 0
-1 + -6x + 3x2 = 0 + 6x + -6x
-1 + -6x + 3x2 = 6x + -6x

Combine like terms: 6x + -6x = 0
-1 + -6x + 3x2 = 0

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

Divide each side by '3'.
-0.3333333333 + -2x + x2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -2x + 0.3333333333 + x2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -2x + x2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -2x + x2 = 0 + 0.3333333333
-2x + x2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-2x + x2 = 0.3333333333

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 = 0.3333333333 + 1

Reorder the terms:
1 + -2x + x2 = 0.3333333333 + 1

Combine like terms: 0.3333333333 + 1 = 1.3333333333
1 + -2x + x2 = 1.3333333333

Factor a perfect square on the left side:
(x + -1)(x + -1) = 1.3333333333

Calculate the square root of the right side: 1.154700538

Break this problem into two subproblems by setting 
(x + -1) equal to 1.154700538 and -1.154700538.

Subproblem 1

x + -1 = 1.154700538 Simplifying x + -1 = 1.154700538 Reorder the terms: -1 + x = 1.154700538 Solving -1 + x = 1.154700538 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 = 1.154700538 + 1 Combine like terms: -1 + 1 = 0 0 + x = 1.154700538 + 1 x = 1.154700538 + 1 Combine like terms: 1.154700538 + 1 = 2.154700538 x = 2.154700538 Simplifying x = 2.154700538

Subproblem 2

x + -1 = -1.154700538 Simplifying x + -1 = -1.154700538 Reorder the terms: -1 + x = -1.154700538 Solving -1 + x = -1.154700538 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 = -1.154700538 + 1 Combine like terms: -1 + 1 = 0 0 + x = -1.154700538 + 1 x = -1.154700538 + 1 Combine like terms: -1.154700538 + 1 = -0.154700538 x = -0.154700538 Simplifying x = -0.154700538

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.154700538, -0.154700538}

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