45x^2-15x-7=0

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Solution for 45x^2-15x-7=0 equation: 


Simplifying
45x2 + -15x + -7 = 0

Reorder the terms:
-7 + -15x + 45x2 = 0

Solving
-7 + -15x + 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'.
-0.1555555556 + -0.3333333333x + x2 = 0

Move the constant term to the right:

Add '0.1555555556' to each side of the equation.
-0.1555555556 + -0.3333333333x + 0.1555555556 + x2 = 0 + 0.1555555556

Reorder the terms:
-0.1555555556 + 0.1555555556 + -0.3333333333x + x2 = 0 + 0.1555555556

Combine like terms: -0.1555555556 + 0.1555555556 = 0.0000000000
0.0000000000 + -0.3333333333x + x2 = 0 + 0.1555555556
-0.3333333333x + x2 = 0 + 0.1555555556

Combine like terms: 0 + 0.1555555556 = 0.1555555556
-0.3333333333x + x2 = 0.1555555556

The x term is -0.3333333333x.  Take half its coefficient (-0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
-0.3333333333x + 0.02777777779 + x2 = 0.1555555556 + 0.02777777779

Reorder the terms:
0.02777777779 + -0.3333333333x + x2 = 0.1555555556 + 0.02777777779

Combine like terms: 0.1555555556 + 0.02777777779 = 0.18333333339
0.02777777779 + -0.3333333333x + x2 = 0.18333333339

Factor a perfect square on the left side:
(x + -0.1666666667)(x + -0.1666666667) = 0.18333333339

Calculate the square root of the right side: 0.428174419

Break this problem into two subproblems by setting 
(x + -0.1666666667) equal to 0.428174419 and -0.428174419.


Subproblem 1
x + -0.1666666667 = 0.428174419

Simplifying
x + -0.1666666667 = 0.428174419

Reorder the terms:
-0.1666666667 + x = 0.428174419

Solving
-0.1666666667 + x = 0.428174419

Solving for variable 'x'.

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

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 0.1666666667 + x = 0.428174419 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + x = 0.428174419 + 0.1666666667
x = 0.428174419 + 0.1666666667

Combine like terms: 0.428174419 + 0.1666666667 = 0.5948410857
x = 0.5948410857

Simplifying
x = 0.5948410857


Subproblem 2
x + -0.1666666667 = -0.428174419

Simplifying
x + -0.1666666667 = -0.428174419

Reorder the terms:
-0.1666666667 + x = -0.428174419

Solving
-0.1666666667 + x = -0.428174419

Solving for variable 'x'.

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

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 0.1666666667 + x = -0.428174419 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + x = -0.428174419 + 0.1666666667
x = -0.428174419 + 0.1666666667

Combine like terms: -0.428174419 + 0.1666666667 = -0.2615077523
x = -0.2615077523

Simplifying
x = -0.2615077523


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.5948410857, -0.2615077523}

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