45x^2-6x-5=0

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Solution for 45x^2-6x-5=0 equation: 


Simplifying
45x2 + -6x + -5 = 0

Reorder the terms:
-5 + -6x + 45x2 = 0

Solving
-5 + -6x + 45x2 = 0

Solving for variable 'x'.

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

Divide each side by '45'.
-0.1111111111 + -0.1333333333x + x2 = 0

Move the constant term to the right:

Add '0.1111111111' to each side of the equation.
-0.1111111111 + -0.1333333333x + 0.1111111111 + x2 = 0 + 0.1111111111

Reorder the terms:
-0.1111111111 + 0.1111111111 + -0.1333333333x + x2 = 0 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + -0.1333333333x + x2 = 0 + 0.1111111111
-0.1333333333x + x2 = 0 + 0.1111111111

Combine like terms: 0 + 0.1111111111 = 0.1111111111
-0.1333333333x + x2 = 0.1111111111

The x term is -0.1333333333x.  Take half its coefficient (-0.06666666665).
Square it (0.004444444442) and add it to both sides.

Add '0.004444444442' to each side of the equation.
-0.1333333333x + 0.004444444442 + x2 = 0.1111111111 + 0.004444444442

Reorder the terms:
0.004444444442 + -0.1333333333x + x2 = 0.1111111111 + 0.004444444442

Combine like terms: 0.1111111111 + 0.004444444442 = 0.115555555542
0.004444444442 + -0.1333333333x + x2 = 0.115555555542

Factor a perfect square on the left side:
(x + -0.06666666665)(x + -0.06666666665) = 0.115555555542

Calculate the square root of the right side: 0.339934634

Break this problem into two subproblems by setting 
(x + -0.06666666665) equal to 0.339934634 and -0.339934634.


Subproblem 1
x + -0.06666666665 = 0.339934634

Simplifying
x + -0.06666666665 = 0.339934634

Reorder the terms:
-0.06666666665 + x = 0.339934634

Solving
-0.06666666665 + x = 0.339934634

Solving for variable 'x'.

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

Add '0.06666666665' to each side of the equation.
-0.06666666665 + 0.06666666665 + x = 0.339934634 + 0.06666666665

Combine like terms: -0.06666666665 + 0.06666666665 = 0.00000000000
0.00000000000 + x = 0.339934634 + 0.06666666665
x = 0.339934634 + 0.06666666665

Combine like terms: 0.339934634 + 0.06666666665 = 0.40660130065
x = 0.40660130065

Simplifying
x = 0.40660130065


Subproblem 2
x + -0.06666666665 = -0.339934634

Simplifying
x + -0.06666666665 = -0.339934634

Reorder the terms:
-0.06666666665 + x = -0.339934634

Solving
-0.06666666665 + x = -0.339934634

Solving for variable 'x'.

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

Add '0.06666666665' to each side of the equation.
-0.06666666665 + 0.06666666665 + x = -0.339934634 + 0.06666666665

Combine like terms: -0.06666666665 + 0.06666666665 = 0.00000000000
0.00000000000 + x = -0.339934634 + 0.06666666665
x = -0.339934634 + 0.06666666665

Combine like terms: -0.339934634 + 0.06666666665 = -0.27326796735
x = -0.27326796735

Simplifying
x = -0.27326796735


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.40660130065, -0.27326796735}

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