4x^2+54x-180=0

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Solution for 4x^2+54x-180=0 equation: 


Simplifying
4x2 + 54x + -180 = 0

Reorder the terms:
-180 + 54x + 4x2 = 0

Solving
-180 + 54x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-90 + 27x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-90 + 27x + 2x2)' equal to zero and attempt to solve:

Simplifying
-90 + 27x + 2x2 = 0

Solving
-90 + 27x + 2x2 = 0

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

Divide each side by '2'.
-45 + 13.5x + x2 = 0

Move the constant term to the right:

Add '45' to each side of the equation.
-45 + 13.5x + 45 + x2 = 0 + 45

Reorder the terms:
-45 + 45 + 13.5x + x2 = 0 + 45

Combine like terms: -45 + 45 = 0
0 + 13.5x + x2 = 0 + 45
13.5x + x2 = 0 + 45

Combine like terms: 0 + 45 = 45
13.5x + x2 = 45

The x term is 13.5x.  Take half its coefficient (6.75).
Square it (45.5625) and add it to both sides.

Add '45.5625' to each side of the equation.
13.5x + 45.5625 + x2 = 45 + 45.5625

Reorder the terms:
45.5625 + 13.5x + x2 = 45 + 45.5625

Combine like terms: 45 + 45.5625 = 90.5625
45.5625 + 13.5x + x2 = 90.5625

Factor a perfect square on the left side:
(x + 6.75)(x + 6.75) = 90.5625

Calculate the square root of the right side: 9.516433155

Break this problem into two subproblems by setting 
(x + 6.75) equal to 9.516433155 and -9.516433155.


Subproblem 1
x + 6.75 = 9.516433155

Simplifying
x + 6.75 = 9.516433155

Reorder the terms:
6.75 + x = 9.516433155

Solving
6.75 + x = 9.516433155

Solving for variable 'x'.

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

Add '-6.75' to each side of the equation.
6.75 + -6.75 + x = 9.516433155 + -6.75

Combine like terms: 6.75 + -6.75 = 0.00
0.00 + x = 9.516433155 + -6.75
x = 9.516433155 + -6.75

Combine like terms: 9.516433155 + -6.75 = 2.766433155
x = 2.766433155

Simplifying
x = 2.766433155


Subproblem 2
x + 6.75 = -9.516433155

Simplifying
x + 6.75 = -9.516433155

Reorder the terms:
6.75 + x = -9.516433155

Solving
6.75 + x = -9.516433155

Solving for variable 'x'.

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

Add '-6.75' to each side of the equation.
6.75 + -6.75 + x = -9.516433155 + -6.75

Combine like terms: 6.75 + -6.75 = 0.00
0.00 + x = -9.516433155 + -6.75
x = -9.516433155 + -6.75

Combine like terms: -9.516433155 + -6.75 = -16.266433155
x = -16.266433155

Simplifying
x = -16.266433155


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.766433155, -16.266433155}
Solution
x = {2.766433155, -16.266433155}

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