-4.5x^2+10x+1.5=0

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Solution for -4.5x^2+10x+1.5=0 equation: 


Simplifying
-4.5x2 + 10x + 1.5 = 0

Reorder the terms:
1.5 + 10x + -4.5x2 = 0

Solving
1.5 + 10x + -4.5x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
-4.5 the coefficient of the squared term: 

Divide each side by '-4.5'.
-0.3333333333 + -2.222222222x + x2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -2.222222222x + 0.3333333333 + x2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -2.222222222x + x2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -2.222222222x + x2 = 0 + 0.3333333333
-2.222222222x + x2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-2.222222222x + x2 = 0.3333333333

The x term is -2.222222222x.  Take half its coefficient (-1.111111111).
Square it (1.234567901) and add it to both sides.

Add '1.234567901' to each side of the equation.
-2.222222222x + 1.234567901 + x2 = 0.3333333333 + 1.234567901

Reorder the terms:
1.234567901 + -2.222222222x + x2 = 0.3333333333 + 1.234567901

Combine like terms: 0.3333333333 + 1.234567901 = 1.5679012343
1.234567901 + -2.222222222x + x2 = 1.5679012343

Factor a perfect square on the left side:
(x + -1.111111111)(x + -1.111111111) = 1.5679012343

Calculate the square root of the right side: 1.25215863

Break this problem into two subproblems by setting 
(x + -1.111111111) equal to 1.25215863 and -1.25215863.


Subproblem 1
x + -1.111111111 = 1.25215863

Simplifying
x + -1.111111111 = 1.25215863

Reorder the terms:
-1.111111111 + x = 1.25215863

Solving
-1.111111111 + x = 1.25215863

Solving for variable 'x'.

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

Add '1.111111111' to each side of the equation.
-1.111111111 + 1.111111111 + x = 1.25215863 + 1.111111111

Combine like terms: -1.111111111 + 1.111111111 = 0.000000000
0.000000000 + x = 1.25215863 + 1.111111111
x = 1.25215863 + 1.111111111

Combine like terms: 1.25215863 + 1.111111111 = 2.363269741
x = 2.363269741

Simplifying
x = 2.363269741


Subproblem 2
x + -1.111111111 = -1.25215863

Simplifying
x + -1.111111111 = -1.25215863

Reorder the terms:
-1.111111111 + x = -1.25215863

Solving
-1.111111111 + x = -1.25215863

Solving for variable 'x'.

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

Add '1.111111111' to each side of the equation.
-1.111111111 + 1.111111111 + x = -1.25215863 + 1.111111111

Combine like terms: -1.111111111 + 1.111111111 = 0.000000000
0.000000000 + x = -1.25215863 + 1.111111111
x = -1.25215863 + 1.111111111

Combine like terms: -1.25215863 + 1.111111111 = -0.141047519
x = -0.141047519

Simplifying
x = -0.141047519


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.363269741, -0.141047519}

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