15x^2-75x+20=0

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Solution for 15x^2-75x+20=0 equation: 


Simplifying
15x2 + -75x + 20 = 0

Reorder the terms:
20 + -75x + 15x2 = 0

Solving
20 + -75x + 15x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '5'.
5(4 + -15x + 3x2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(4 + -15x + 3x2)' equal to zero and attempt to solve:

Simplifying
4 + -15x + 3x2 = 0

Solving
4 + -15x + 3x2 = 0

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

Divide each side by '3'.
1.333333333 + -5x + x2 = 0

Move the constant term to the right:

Add '-1.333333333' to each side of the equation.
1.333333333 + -5x + -1.333333333 + x2 = 0 + -1.333333333

Reorder the terms:
1.333333333 + -1.333333333 + -5x + x2 = 0 + -1.333333333

Combine like terms: 1.333333333 + -1.333333333 = 0.000000000
0.000000000 + -5x + x2 = 0 + -1.333333333
-5x + x2 = 0 + -1.333333333

Combine like terms: 0 + -1.333333333 = -1.333333333
-5x + x2 = -1.333333333

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = -1.333333333 + 6.25

Reorder the terms:
6.25 + -5x + x2 = -1.333333333 + 6.25

Combine like terms: -1.333333333 + 6.25 = 4.916666667
6.25 + -5x + x2 = 4.916666667

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 4.916666667

Calculate the square root of the right side: 2.217355783

Break this problem into two subproblems by setting 
(x + -2.5) equal to 2.217355783 and -2.217355783.


Subproblem 1
x + -2.5 = 2.217355783

Simplifying
x + -2.5 = 2.217355783

Reorder the terms:
-2.5 + x = 2.217355783

Solving
-2.5 + x = 2.217355783

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = 2.217355783 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = 2.217355783 + 2.5
x = 2.217355783 + 2.5

Combine like terms: 2.217355783 + 2.5 = 4.717355783
x = 4.717355783

Simplifying
x = 4.717355783


Subproblem 2
x + -2.5 = -2.217355783

Simplifying
x + -2.5 = -2.217355783

Reorder the terms:
-2.5 + x = -2.217355783

Solving
-2.5 + x = -2.217355783

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = -2.217355783 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = -2.217355783 + 2.5
x = -2.217355783 + 2.5

Combine like terms: -2.217355783 + 2.5 = 0.282644217
x = 0.282644217

Simplifying
x = 0.282644217


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.717355783, 0.282644217}
Solution
x = {4.717355783, 0.282644217}

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