10x^2-90x+145=0

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Solution for 10x^2-90x+145=0 equation:


Simplifying
10x2 + -90x + 145 = 0

Reorder the terms:
145 + -90x + 10x2 = 0

Solving
145 + -90x + 10x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '5'.
5(29 + -18x + 2x2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(29 + -18x + 2x2)' equal to zero and attempt to solve:

Simplifying
29 + -18x + 2x2 = 0

Solving
29 + -18x + 2x2 = 0

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

Divide each side by '2'.
14.5 + -9x + x2 = 0

Move the constant term to the right:

Add '-14.5' to each side of the equation.
14.5 + -9x + -14.5 + x2 = 0 + -14.5

Reorder the terms:
14.5 + -14.5 + -9x + x2 = 0 + -14.5

Combine like terms: 14.5 + -14.5 = 0.0
0.0 + -9x + x2 = 0 + -14.5
-9x + x2 = 0 + -14.5

Combine like terms: 0 + -14.5 = -14.5
-9x + x2 = -14.5

The x term is -9x.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9x + 20.25 + x2 = -14.5 + 20.25

Reorder the terms:
20.25 + -9x + x2 = -14.5 + 20.25

Combine like terms: -14.5 + 20.25 = 5.75
20.25 + -9x + x2 = 5.75

Factor a perfect square on the left side:
(x + -4.5)(x + -4.5) = 5.75

Calculate the square root of the right side: 2.397915762

Break this problem into two subproblems by setting 
(x + -4.5) equal to 2.397915762 and -2.397915762.


Subproblem 1
x + -4.5 = 2.397915762

Simplifying
x + -4.5 = 2.397915762

Reorder the terms:
-4.5 + x = 2.397915762

Solving
-4.5 + x = 2.397915762

Solving for variable 'x'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + x = 2.397915762 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + x = 2.397915762 + 4.5
x = 2.397915762 + 4.5

Combine like terms: 2.397915762 + 4.5 = 6.897915762
x = 6.897915762

Simplifying
x = 6.897915762


Subproblem 2
x + -4.5 = -2.397915762

Simplifying
x + -4.5 = -2.397915762

Reorder the terms:
-4.5 + x = -2.397915762

Solving
-4.5 + x = -2.397915762

Solving for variable 'x'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + x = -2.397915762 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + x = -2.397915762 + 4.5
x = -2.397915762 + 4.5

Combine like terms: -2.397915762 + 4.5 = 2.102084238
x = 2.102084238

Simplifying
x = 2.102084238


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.897915762, 2.102084238}
Solution
x = {6.897915762, 2.102084238}

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