4x+3x^2=180

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Solution for 4x+3x^2=180 equation:


Simplifying
4x + 3x2 = 180

Solving
4x + 3x2 = 180

Solving for variable 'x'.

Reorder the terms:
-180 + 4x + 3x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + 4x + 3x2 = 0

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

Divide each side by '3'.
-60 + 1.333333333x + x2 = 0

Move the constant term to the right:

Add '60' to each side of the equation.
-60 + 1.333333333x + 60 + x2 = 0 + 60

Reorder the terms:
-60 + 60 + 1.333333333x + x2 = 0 + 60

Combine like terms: -60 + 60 = 0
0 + 1.333333333x + x2 = 0 + 60
1.333333333x + x2 = 0 + 60

Combine like terms: 0 + 60 = 60
1.333333333x + x2 = 60

The x term is 1.333333333x.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333x + 0.4444444442 + x2 = 60 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333x + x2 = 60 + 0.4444444442

Combine like terms: 60 + 0.4444444442 = 60.4444444442
0.4444444442 + 1.333333333x + x2 = 60.4444444442

Factor a perfect square on the left side:
(x + 0.6666666665)(x + 0.6666666665) = 60.4444444442

Calculate the square root of the right side: 7.774602526

Break this problem into two subproblems by setting 
(x + 0.6666666665) equal to 7.774602526 and -7.774602526.

Subproblem 1

x + 0.6666666665 = 7.774602526 Simplifying x + 0.6666666665 = 7.774602526 Reorder the terms: 0.6666666665 + x = 7.774602526 Solving 0.6666666665 + x = 7.774602526 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = 7.774602526 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = 7.774602526 + -0.6666666665 x = 7.774602526 + -0.6666666665 Combine like terms: 7.774602526 + -0.6666666665 = 7.1079358595 x = 7.1079358595 Simplifying x = 7.1079358595

Subproblem 2

x + 0.6666666665 = -7.774602526 Simplifying x + 0.6666666665 = -7.774602526 Reorder the terms: 0.6666666665 + x = -7.774602526 Solving 0.6666666665 + x = -7.774602526 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = -7.774602526 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = -7.774602526 + -0.6666666665 x = -7.774602526 + -0.6666666665 Combine like terms: -7.774602526 + -0.6666666665 = -8.4412691925 x = -8.4412691925 Simplifying x = -8.4412691925

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.1079358595, -8.4412691925}

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