-10x^2-30x+22.5=0

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Solution for -10x^2-30x+22.5=0 equation: 


Simplifying
-10x2 + -30x + 22.5 = 0

Reorder the terms:
22.5 + -30x + -10x2 = 0

Solving
22.5 + -30x + -10x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
-10 the coefficient of the squared term: 

Divide each side by '-10'.
-2.25 + 3x + x2 = 0

Move the constant term to the right:

Add '2.25' to each side of the equation.
-2.25 + 3x + 2.25 + x2 = 0 + 2.25

Reorder the terms:
-2.25 + 2.25 + 3x + x2 = 0 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + 3x + x2 = 0 + 2.25
3x + x2 = 0 + 2.25

Combine like terms: 0 + 2.25 = 2.25
3x + x2 = 2.25

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 = 2.25 + 2.25

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

Combine like terms: 2.25 + 2.25 = 4.5
2.25 + 3x + x2 = 4.5

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 4.5

Calculate the square root of the right side: 2.121320344

Break this problem into two subproblems by setting 
(x + 1.5) equal to 2.121320344 and -2.121320344.


Subproblem 1
x + 1.5 = 2.121320344

Simplifying
x + 1.5 = 2.121320344

Reorder the terms:
1.5 + x = 2.121320344

Solving
1.5 + x = 2.121320344

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 = 2.121320344 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 2.121320344 + -1.5
x = 2.121320344 + -1.5

Combine like terms: 2.121320344 + -1.5 = 0.621320344
x = 0.621320344

Simplifying
x = 0.621320344


Subproblem 2
x + 1.5 = -2.121320344

Simplifying
x + 1.5 = -2.121320344

Reorder the terms:
1.5 + x = -2.121320344

Solving
1.5 + x = -2.121320344

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 = -2.121320344 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -2.121320344 + -1.5
x = -2.121320344 + -1.5

Combine like terms: -2.121320344 + -1.5 = -3.621320344
x = -3.621320344

Simplifying
x = -3.621320344


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.621320344, -3.621320344}

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