74x(2x+7)=180

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Solution for 74x(2x+7)=180 equation: 


Simplifying
74x(2x + 7) = 180

Reorder the terms:
74x(7 + 2x) = 180
(7 * 74x + 2x * 74x) = 180
(518x + 148x2) = 180

Solving
518x + 148x2 = 180

Solving for variable 'x'.

Reorder the terms:
-180 + 518x + 148x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + 518x + 148x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-90 + 259x + 74x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-90 + 259x + 74x2)' equal to zero and attempt to solve:

Simplifying
-90 + 259x + 74x2 = 0

Solving
-90 + 259x + 74x2 = 0

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

Divide each side by '74'.
-1.216216216 + 3.5x + x2 = 0

Move the constant term to the right:

Add '1.216216216' to each side of the equation.
-1.216216216 + 3.5x + 1.216216216 + x2 = 0 + 1.216216216

Reorder the terms:
-1.216216216 + 1.216216216 + 3.5x + x2 = 0 + 1.216216216

Combine like terms: -1.216216216 + 1.216216216 = 0.000000000
0.000000000 + 3.5x + x2 = 0 + 1.216216216
3.5x + x2 = 0 + 1.216216216

Combine like terms: 0 + 1.216216216 = 1.216216216
3.5x + x2 = 1.216216216

The x term is 3.5x.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5x + 3.0625 + x2 = 1.216216216 + 3.0625

Reorder the terms:
3.0625 + 3.5x + x2 = 1.216216216 + 3.0625

Combine like terms: 1.216216216 + 3.0625 = 4.278716216
3.0625 + 3.5x + x2 = 4.278716216

Factor a perfect square on the left side:
(x + 1.75)(x + 1.75) = 4.278716216

Calculate the square root of the right side: 2.068505793

Break this problem into two subproblems by setting 
(x + 1.75) equal to 2.068505793 and -2.068505793.


Subproblem 1
x + 1.75 = 2.068505793

Simplifying
x + 1.75 = 2.068505793

Reorder the terms:
1.75 + x = 2.068505793

Solving
1.75 + x = 2.068505793

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.75' to each side of the equation.
1.75 + -1.75 + x = 2.068505793 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + x = 2.068505793 + -1.75
x = 2.068505793 + -1.75

Combine like terms: 2.068505793 + -1.75 = 0.318505793
x = 0.318505793

Simplifying
x = 0.318505793


Subproblem 2
x + 1.75 = -2.068505793

Simplifying
x + 1.75 = -2.068505793

Reorder the terms:
1.75 + x = -2.068505793

Solving
1.75 + x = -2.068505793

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.75' to each side of the equation.
1.75 + -1.75 + x = -2.068505793 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + x = -2.068505793 + -1.75
x = -2.068505793 + -1.75

Combine like terms: -2.068505793 + -1.75 = -3.818505793
x = -3.818505793

Simplifying
x = -3.818505793


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.318505793, -3.818505793}
Solution
x = {0.318505793, -3.818505793}

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