1.35x^2-8.4x+4.5=0

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Solution for 1.35x^2-8.4x+4.5=0 equation: 


Simplifying
1.35x2 + -8.4x + 4.5 = 0

Reorder the terms:
4.5 + -8.4x + 1.35x2 = 0

Solving
4.5 + -8.4x + 1.35x2 = 0

Solving for variable 'x'.

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

Divide each side by '1.35'.
3.333333333 + -6.222222222x + x2 = 0

Move the constant term to the right:

Add '-3.333333333' to each side of the equation.
3.333333333 + -6.222222222x + -3.333333333 + x2 = 0 + -3.333333333

Reorder the terms:
3.333333333 + -3.333333333 + -6.222222222x + x2 = 0 + -3.333333333

Combine like terms: 3.333333333 + -3.333333333 = 0.000000000
0.000000000 + -6.222222222x + x2 = 0 + -3.333333333
-6.222222222x + x2 = 0 + -3.333333333

Combine like terms: 0 + -3.333333333 = -3.333333333
-6.222222222x + x2 = -3.333333333

The x term is -6.222222222x.  Take half its coefficient (-3.111111111).
Square it (9.679012345) and add it to both sides.

Add '9.679012345' to each side of the equation.
-6.222222222x + 9.679012345 + x2 = -3.333333333 + 9.679012345

Reorder the terms:
9.679012345 + -6.222222222x + x2 = -3.333333333 + 9.679012345

Combine like terms: -3.333333333 + 9.679012345 = 6.345679012
9.679012345 + -6.222222222x + x2 = 6.345679012

Factor a perfect square on the left side:
(x + -3.111111111)(x + -3.111111111) = 6.345679012

Calculate the square root of the right side: 2.519063122

Break this problem into two subproblems by setting 
(x + -3.111111111) equal to 2.519063122 and -2.519063122.


Subproblem 1
x + -3.111111111 = 2.519063122

Simplifying
x + -3.111111111 = 2.519063122

Reorder the terms:
-3.111111111 + x = 2.519063122

Solving
-3.111111111 + x = 2.519063122

Solving for variable 'x'.

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

Add '3.111111111' to each side of the equation.
-3.111111111 + 3.111111111 + x = 2.519063122 + 3.111111111

Combine like terms: -3.111111111 + 3.111111111 = 0.000000000
0.000000000 + x = 2.519063122 + 3.111111111
x = 2.519063122 + 3.111111111

Combine like terms: 2.519063122 + 3.111111111 = 5.630174233
x = 5.630174233

Simplifying
x = 5.630174233


Subproblem 2
x + -3.111111111 = -2.519063122

Simplifying
x + -3.111111111 = -2.519063122

Reorder the terms:
-3.111111111 + x = -2.519063122

Solving
-3.111111111 + x = -2.519063122

Solving for variable 'x'.

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

Add '3.111111111' to each side of the equation.
-3.111111111 + 3.111111111 + x = -2.519063122 + 3.111111111

Combine like terms: -3.111111111 + 3.111111111 = 0.000000000
0.000000000 + x = -2.519063122 + 3.111111111
x = -2.519063122 + 3.111111111

Combine like terms: -2.519063122 + 3.111111111 = 0.592047989
x = 0.592047989

Simplifying
x = 0.592047989


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.630174233, 0.592047989}

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