3x^2+x=111

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Solution for 3x^2+x=111 equation:


Simplifying
3x2 + x = 111

Reorder the terms:
x + 3x2 = 111

Solving
x + 3x2 = 111

Solving for variable 'x'.

Reorder the terms:
-111 + x + 3x2 = 111 + -111

Combine like terms: 111 + -111 = 0
-111 + x + 3x2 = 0

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

Divide each side by '3'.
-37 + 0.3333333333x + x2 = 0

Move the constant term to the right:

Add '37' to each side of the equation.
-37 + 0.3333333333x + 37 + x2 = 0 + 37

Reorder the terms:
-37 + 37 + 0.3333333333x + x2 = 0 + 37

Combine like terms: -37 + 37 = 0
0 + 0.3333333333x + x2 = 0 + 37
0.3333333333x + x2 = 0 + 37

Combine like terms: 0 + 37 = 37
0.3333333333x + x2 = 37

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 = 37 + 0.25

Reorder the terms:
0.25 + 0.3333333333x + x2 = 37 + 0.25

Combine like terms: 37 + 0.25 = 37.25
0.25 + 0.3333333333x + x2 = 37.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 37.25

Calculate the square root of the right side: 6.103277808

Break this problem into two subproblems by setting 
(x + 0.5) equal to 6.103277808 and -6.103277808.

Subproblem 1

x + 0.5 = 6.103277808 Simplifying x + 0.5 = 6.103277808 Reorder the terms: 0.5 + x = 6.103277808 Solving 0.5 + x = 6.103277808 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 = 6.103277808 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 6.103277808 + -0.5 x = 6.103277808 + -0.5 Combine like terms: 6.103277808 + -0.5 = 5.603277808 x = 5.603277808 Simplifying x = 5.603277808

Subproblem 2

x + 0.5 = -6.103277808 Simplifying x + 0.5 = -6.103277808 Reorder the terms: 0.5 + x = -6.103277808 Solving 0.5 + x = -6.103277808 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 = -6.103277808 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -6.103277808 + -0.5 x = -6.103277808 + -0.5 Combine like terms: -6.103277808 + -0.5 = -6.603277808 x = -6.603277808 Simplifying x = -6.603277808

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.603277808, -6.603277808}

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