20x(x-3)=-5

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Solution for 20x(x-3)=-5 equation: 


Simplifying
20x(x + -3) = -5

Reorder the terms:
20x(-3 + x) = -5
(-3 * 20x + x * 20x) = -5
(-60x + 20x2) = -5

Solving
-60x + 20x2 = -5

Solving for variable 'x'.

Reorder the terms:
5 + -60x + 20x2 = -5 + 5

Combine like terms: -5 + 5 = 0
5 + -60x + 20x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(1 + -12x + 4x2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(1 + -12x + 4x2)' equal to zero and attempt to solve:

Simplifying
1 + -12x + 4x2 = 0

Solving
1 + -12x + 4x2 = 0

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

Divide each side by '4'.
0.25 + -3x + x2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + -3x + -0.25 + x2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + -3x + x2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + -3x + x2 = 0 + -0.25
-3x + x2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
-3x + x2 = -0.25

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = -0.25 + 2.25

Reorder the terms:
2.25 + -3x + x2 = -0.25 + 2.25

Combine like terms: -0.25 + 2.25 = 2
2.25 + -3x + x2 = 2

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x + -1.5) equal to 1.414213562 and -1.414213562.


Subproblem 1
x + -1.5 = 1.414213562

Simplifying
x + -1.5 = 1.414213562

Reorder the terms:
-1.5 + x = 1.414213562

Solving
-1.5 + x = 1.414213562

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = 1.414213562 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 1.414213562 + 1.5
x = 1.414213562 + 1.5

Combine like terms: 1.414213562 + 1.5 = 2.914213562
x = 2.914213562

Simplifying
x = 2.914213562


Subproblem 2
x + -1.5 = -1.414213562

Simplifying
x + -1.5 = -1.414213562

Reorder the terms:
-1.5 + x = -1.414213562

Solving
-1.5 + x = -1.414213562

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = -1.414213562 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -1.414213562 + 1.5
x = -1.414213562 + 1.5

Combine like terms: -1.414213562 + 1.5 = 0.085786438
x = 0.085786438

Simplifying
x = 0.085786438


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.914213562, 0.085786438}
Solution
x = {2.914213562, 0.085786438}

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