6x^2+15x+75=180

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Solution for 6x^2+15x+75=180 equation: 


Simplifying
6x2 + 15x + 75 = 180

Reorder the terms:
75 + 15x + 6x2 = 180

Solving
75 + 15x + 6x2 = 180

Solving for variable 'x'.

Reorder the terms:
75 + -180 + 15x + 6x2 = 180 + -180

Combine like terms: 75 + -180 = -105
-105 + 15x + 6x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-105 + 15x + 6x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-35 + 5x + 2x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-35 + 5x + 2x2)' equal to zero and attempt to solve:

Simplifying
-35 + 5x + 2x2 = 0

Solving
-35 + 5x + 2x2 = 0

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

Divide each side by '2'.
-17.5 + 2.5x + x2 = 0

Move the constant term to the right:

Add '17.5' to each side of the equation.
-17.5 + 2.5x + 17.5 + x2 = 0 + 17.5

Reorder the terms:
-17.5 + 17.5 + 2.5x + x2 = 0 + 17.5

Combine like terms: -17.5 + 17.5 = 0.0
0.0 + 2.5x + x2 = 0 + 17.5
2.5x + x2 = 0 + 17.5

Combine like terms: 0 + 17.5 = 17.5
2.5x + x2 = 17.5

The x term is 2.5x.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5x + 1.5625 + x2 = 17.5 + 1.5625

Reorder the terms:
1.5625 + 2.5x + x2 = 17.5 + 1.5625

Combine like terms: 17.5 + 1.5625 = 19.0625
1.5625 + 2.5x + x2 = 19.0625

Factor a perfect square on the left side:
(x + 1.25)(x + 1.25) = 19.0625

Calculate the square root of the right side: 4.366062299

Break this problem into two subproblems by setting 
(x + 1.25) equal to 4.366062299 and -4.366062299.


Subproblem 1
x + 1.25 = 4.366062299

Simplifying
x + 1.25 = 4.366062299

Reorder the terms:
1.25 + x = 4.366062299

Solving
1.25 + x = 4.366062299

Solving for variable 'x'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + x = 4.366062299 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + x = 4.366062299 + -1.25
x = 4.366062299 + -1.25

Combine like terms: 4.366062299 + -1.25 = 3.116062299
x = 3.116062299

Simplifying
x = 3.116062299


Subproblem 2
x + 1.25 = -4.366062299

Simplifying
x + 1.25 = -4.366062299

Reorder the terms:
1.25 + x = -4.366062299

Solving
1.25 + x = -4.366062299

Solving for variable 'x'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + x = -4.366062299 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + x = -4.366062299 + -1.25
x = -4.366062299 + -1.25

Combine like terms: -4.366062299 + -1.25 = -5.616062299
x = -5.616062299

Simplifying
x = -5.616062299


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.116062299, -5.616062299}
Solution
x = {3.116062299, -5.616062299}

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