3x^2+x+4=-x+13

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Solution for 3x^2+x+4=-x+13 equation:


Simplifying
3x2 + x + 4 = -1x + 13

Reorder the terms:
4 + x + 3x2 = -1x + 13

Reorder the terms:
4 + x + 3x2 = 13 + -1x

Solving
4 + x + 3x2 = 13 + -1x

Solving for variable 'x'.

Reorder the terms:
4 + -13 + x + x + 3x2 = 13 + -1x + -13 + x

Combine like terms: 4 + -13 = -9
-9 + x + x + 3x2 = 13 + -1x + -13 + x

Combine like terms: x + x = 2x
-9 + 2x + 3x2 = 13 + -1x + -13 + x

Reorder the terms:
-9 + 2x + 3x2 = 13 + -13 + -1x + x

Combine like terms: 13 + -13 = 0
-9 + 2x + 3x2 = 0 + -1x + x
-9 + 2x + 3x2 = -1x + x

Combine like terms: -1x + x = 0
-9 + 2x + 3x2 = 0

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

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

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 0.6666666667x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + 0.6666666667x + x2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 0.6666666667x + x2 = 0 + 3
0.6666666667x + x2 = 0 + 3

Combine like terms: 0 + 3 = 3
0.6666666667x + x2 = 3

The x term is 0.6666666667x.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667x + 0.1111111112 + x2 = 3 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667x + x2 = 3 + 0.1111111112

Combine like terms: 3 + 0.1111111112 = 3.1111111112
0.1111111112 + 0.6666666667x + x2 = 3.1111111112

Factor a perfect square on the left side:
(x + 0.3333333334)(x + 0.3333333334) = 3.1111111112

Calculate the square root of the right side: 1.763834207

Break this problem into two subproblems by setting 
(x + 0.3333333334) equal to 1.763834207 and -1.763834207.

Subproblem 1

x + 0.3333333334 = 1.763834207 Simplifying x + 0.3333333334 = 1.763834207 Reorder the terms: 0.3333333334 + x = 1.763834207 Solving 0.3333333334 + x = 1.763834207 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = 1.763834207 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = 1.763834207 + -0.3333333334 x = 1.763834207 + -0.3333333334 Combine like terms: 1.763834207 + -0.3333333334 = 1.4305008736 x = 1.4305008736 Simplifying x = 1.4305008736

Subproblem 2

x + 0.3333333334 = -1.763834207 Simplifying x + 0.3333333334 = -1.763834207 Reorder the terms: 0.3333333334 + x = -1.763834207 Solving 0.3333333334 + x = -1.763834207 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = -1.763834207 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = -1.763834207 + -0.3333333334 x = -1.763834207 + -0.3333333334 Combine like terms: -1.763834207 + -0.3333333334 = -2.0971675404 x = -2.0971675404 Simplifying x = -2.0971675404

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.4305008736, -2.0971675404}

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