x+10=3x^2+1

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Solution for x+10=3x^2+1 equation:


Simplifying
x + 10 = 3x2 + 1

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

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

Solving
10 + x = 1 + 3x2

Solving for variable 'x'.

Reorder the terms:
10 + -1 + x + -3x2 = 1 + 3x2 + -1 + -3x2

Combine like terms: 10 + -1 = 9
9 + x + -3x2 = 1 + 3x2 + -1 + -3x2

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

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

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

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

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

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -0.3333333333x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + -0.3333333333x + x2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -0.3333333333x + x2 = 0 + 3
-0.3333333333x + x2 = 0 + 3

Combine like terms: 0 + 3 = 3
-0.3333333333x + x2 = 3

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.
-0.3333333333x + 0.25 + x2 = 3 + 0.25

Reorder the terms:
0.25 + -0.3333333333x + x2 = 3 + 0.25

Combine like terms: 3 + 0.25 = 3.25
0.25 + -0.3333333333x + x2 = 3.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(x + 0.5) equal to 1.802775638 and -1.802775638.

Subproblem 1

x + 0.5 = 1.802775638 Simplifying x + 0.5 = 1.802775638 Reorder the terms: 0.5 + x = 1.802775638 Solving 0.5 + x = 1.802775638 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 = 1.802775638 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1.802775638 + -0.5 x = 1.802775638 + -0.5 Combine like terms: 1.802775638 + -0.5 = 1.302775638 x = 1.302775638 Simplifying x = 1.302775638

Subproblem 2

x + 0.5 = -1.802775638 Simplifying x + 0.5 = -1.802775638 Reorder the terms: 0.5 + x = -1.802775638 Solving 0.5 + x = -1.802775638 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 = -1.802775638 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1.802775638 + -0.5 x = -1.802775638 + -0.5 Combine like terms: -1.802775638 + -0.5 = -2.302775638 x = -2.302775638 Simplifying x = -2.302775638

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.302775638, -2.302775638}

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