45x^2-42x-5=0

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Solution for 45x^2-42x-5=0 equation: 


Simplifying
45x2 + -42x + -5 = 0

Reorder the terms:
-5 + -42x + 45x2 = 0

Solving
-5 + -42x + 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 + -0.9333333333x + x2 = 0

Move the constant term to the right:

Add '0.1111111111' to each side of the equation.
-0.1111111111 + -0.9333333333x + 0.1111111111 + x2 = 0 + 0.1111111111

Reorder the terms:
-0.1111111111 + 0.1111111111 + -0.9333333333x + x2 = 0 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + -0.9333333333x + x2 = 0 + 0.1111111111
-0.9333333333x + x2 = 0 + 0.1111111111

Combine like terms: 0 + 0.1111111111 = 0.1111111111
-0.9333333333x + x2 = 0.1111111111

The x term is -0.9333333333x.  Take half its coefficient (-0.4666666667).
Square it (0.2177777778) and add it to both sides.

Add '0.2177777778' to each side of the equation.
-0.9333333333x + 0.2177777778 + x2 = 0.1111111111 + 0.2177777778

Reorder the terms:
0.2177777778 + -0.9333333333x + x2 = 0.1111111111 + 0.2177777778

Combine like terms: 0.1111111111 + 0.2177777778 = 0.3288888889
0.2177777778 + -0.9333333333x + x2 = 0.3288888889

Factor a perfect square on the left side:
(x + -0.4666666667)(x + -0.4666666667) = 0.3288888889

Calculate the square root of the right side: 0.573488351

Break this problem into two subproblems by setting 
(x + -0.4666666667) equal to 0.573488351 and -0.573488351.


Subproblem 1
x + -0.4666666667 = 0.573488351

Simplifying
x + -0.4666666667 = 0.573488351

Reorder the terms:
-0.4666666667 + x = 0.573488351

Solving
-0.4666666667 + x = 0.573488351

Solving for variable 'x'.

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

Add '0.4666666667' to each side of the equation.
-0.4666666667 + 0.4666666667 + x = 0.573488351 + 0.4666666667

Combine like terms: -0.4666666667 + 0.4666666667 = 0.0000000000
0.0000000000 + x = 0.573488351 + 0.4666666667
x = 0.573488351 + 0.4666666667

Combine like terms: 0.573488351 + 0.4666666667 = 1.0401550177
x = 1.0401550177

Simplifying
x = 1.0401550177


Subproblem 2
x + -0.4666666667 = -0.573488351

Simplifying
x + -0.4666666667 = -0.573488351

Reorder the terms:
-0.4666666667 + x = -0.573488351

Solving
-0.4666666667 + x = -0.573488351

Solving for variable 'x'.

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

Add '0.4666666667' to each side of the equation.
-0.4666666667 + 0.4666666667 + x = -0.573488351 + 0.4666666667

Combine like terms: -0.4666666667 + 0.4666666667 = 0.0000000000
0.0000000000 + x = -0.573488351 + 0.4666666667
x = -0.573488351 + 0.4666666667

Combine like terms: -0.573488351 + 0.4666666667 = -0.1068216843
x = -0.1068216843

Simplifying
x = -0.1068216843


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.0401550177, -0.1068216843}

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