37+x(3x+51)=180

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Solution for 37+x(3x+51)=180 equation:


Simplifying
37 + x(3x + 51) = 180

Reorder the terms:
37 + x(51 + 3x) = 180
37 + (51 * x + 3x * x) = 180
37 + (51x + 3x2) = 180

Solving
37 + 51x + 3x2 = 180

Solving for variable 'x'.

Reorder the terms:
37 + -180 + 51x + 3x2 = 180 + -180

Combine like terms: 37 + -180 = -143
-143 + 51x + 3x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-143 + 51x + 3x2 = 0

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

Divide each side by '3'.
-47.66666667 + 17x + x2 = 0

Move the constant term to the right:

Add '47.66666667' to each side of the equation.
-47.66666667 + 17x + 47.66666667 + x2 = 0 + 47.66666667

Reorder the terms:
-47.66666667 + 47.66666667 + 17x + x2 = 0 + 47.66666667

Combine like terms: -47.66666667 + 47.66666667 = 0.00000000
0.00000000 + 17x + x2 = 0 + 47.66666667
17x + x2 = 0 + 47.66666667

Combine like terms: 0 + 47.66666667 = 47.66666667
17x + x2 = 47.66666667

The x term is 17x.  Take half its coefficient (8.5).
Square it (72.25) and add it to both sides.

Add '72.25' to each side of the equation.
17x + 72.25 + x2 = 47.66666667 + 72.25

Reorder the terms:
72.25 + 17x + x2 = 47.66666667 + 72.25

Combine like terms: 47.66666667 + 72.25 = 119.91666667
72.25 + 17x + x2 = 119.91666667

Factor a perfect square on the left side:
(x + 8.5)(x + 8.5) = 119.91666667

Calculate the square root of the right side: 10.950646861

Break this problem into two subproblems by setting 
(x + 8.5) equal to 10.950646861 and -10.950646861.

Subproblem 1

x + 8.5 = 10.950646861 Simplifying x + 8.5 = 10.950646861 Reorder the terms: 8.5 + x = 10.950646861 Solving 8.5 + x = 10.950646861 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8.5' to each side of the equation. 8.5 + -8.5 + x = 10.950646861 + -8.5 Combine like terms: 8.5 + -8.5 = 0.0 0.0 + x = 10.950646861 + -8.5 x = 10.950646861 + -8.5 Combine like terms: 10.950646861 + -8.5 = 2.450646861 x = 2.450646861 Simplifying x = 2.450646861

Subproblem 2

x + 8.5 = -10.950646861 Simplifying x + 8.5 = -10.950646861 Reorder the terms: 8.5 + x = -10.950646861 Solving 8.5 + x = -10.950646861 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8.5' to each side of the equation. 8.5 + -8.5 + x = -10.950646861 + -8.5 Combine like terms: 8.5 + -8.5 = 0.0 0.0 + x = -10.950646861 + -8.5 x = -10.950646861 + -8.5 Combine like terms: -10.950646861 + -8.5 = -19.450646861 x = -19.450646861 Simplifying x = -19.450646861

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.450646861, -19.450646861}

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