2x(4x-6)=180

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Solution for 2x(4x-6)=180 equation: 


Simplifying
2x(4x + -6) = 180

Reorder the terms:
2x(-6 + 4x) = 180
(-6 * 2x + 4x * 2x) = 180
(-12x + 8x2) = 180

Solving
-12x + 8x2 = 180

Solving for variable 'x'.

Reorder the terms:
-180 + -12x + 8x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + -12x + 8x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-45 + -3x + 2x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-45 + -3x + 2x2)' equal to zero and attempt to solve:

Simplifying
-45 + -3x + 2x2 = 0

Solving
-45 + -3x + 2x2 = 0

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

Divide each side by '2'.
-22.5 + -1.5x + x2 = 0

Move the constant term to the right:

Add '22.5' to each side of the equation.
-22.5 + -1.5x + 22.5 + x2 = 0 + 22.5

Reorder the terms:
-22.5 + 22.5 + -1.5x + x2 = 0 + 22.5

Combine like terms: -22.5 + 22.5 = 0.0
0.0 + -1.5x + x2 = 0 + 22.5
-1.5x + x2 = 0 + 22.5

Combine like terms: 0 + 22.5 = 22.5
-1.5x + x2 = 22.5

The x term is -1.5x.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5x + 0.5625 + x2 = 22.5 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 22.5 + 0.5625

Combine like terms: 22.5 + 0.5625 = 23.0625
0.5625 + -1.5x + x2 = 23.0625

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 23.0625

Calculate the square root of the right side: 4.802343178

Break this problem into two subproblems by setting 
(x + -0.75) equal to 4.802343178 and -4.802343178.


Subproblem 1
x + -0.75 = 4.802343178

Simplifying
x + -0.75 = 4.802343178

Reorder the terms:
-0.75 + x = 4.802343178

Solving
-0.75 + x = 4.802343178

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = 4.802343178 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = 4.802343178 + 0.75
x = 4.802343178 + 0.75

Combine like terms: 4.802343178 + 0.75 = 5.552343178
x = 5.552343178

Simplifying
x = 5.552343178


Subproblem 2
x + -0.75 = -4.802343178

Simplifying
x + -0.75 = -4.802343178

Reorder the terms:
-0.75 + x = -4.802343178

Solving
-0.75 + x = -4.802343178

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = -4.802343178 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = -4.802343178 + 0.75
x = -4.802343178 + 0.75

Combine like terms: -4.802343178 + 0.75 = -4.052343178
x = -4.052343178

Simplifying
x = -4.052343178


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.552343178, -4.052343178}
Solution
x = {5.552343178, -4.052343178}

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