45x^2-21x-1=0

Solution for 45x^2-21x-1=0 equation:

Simplifying
45x2 + -21x + -1 = 0

Reorder the terms:
-1 + -21x + 45x2 = 0

Solving
-1 + -21x + 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'.
-0.02222222222 + -0.4666666667x + x2 = 0

Move the constant term to the right:

Add '0.02222222222' to each side of the equation.
-0.02222222222 + -0.4666666667x + 0.02222222222 + x2 = 0 + 0.02222222222

Reorder the terms:
-0.02222222222 + 0.02222222222 + -0.4666666667x + x2 = 0 + 0.02222222222

Combine like terms: -0.02222222222 + 0.02222222222 = 0.00000000000
0.00000000000 + -0.4666666667x + x2 = 0 + 0.02222222222
-0.4666666667x + x2 = 0 + 0.02222222222

Combine like terms: 0 + 0.02222222222 = 0.02222222222
-0.4666666667x + x2 = 0.02222222222

The x term is -0.4666666667x.  Take half its coefficient (-0.2333333334).
Square it (0.05444444448) and add it to both sides.

Add '0.05444444448' to each side of the equation.
-0.4666666667x + 0.05444444448 + x2 = 0.02222222222 + 0.05444444448

Reorder the terms:
0.05444444448 + -0.4666666667x + x2 = 0.02222222222 + 0.05444444448

Combine like terms: 0.02222222222 + 0.05444444448 = 0.0766666667
0.05444444448 + -0.4666666667x + x2 = 0.0766666667

Factor a perfect square on the left side:
(x + -0.2333333334)(x + -0.2333333334) = 0.0766666667

Calculate the square root of the right side: 0.276887462

Break this problem into two subproblems by setting 
(x + -0.2333333334) equal to 0.276887462 and -0.276887462.

Subproblem 1
x + -0.2333333334 = 0.276887462

Simplifying
x + -0.2333333334 = 0.276887462

Reorder the terms:
-0.2333333334 + x = 0.276887462

Solving
-0.2333333334 + x = 0.276887462

Solving for variable 'x'.

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

Add '0.2333333334' to each side of the equation.
-0.2333333334 + 0.2333333334 + x = 0.276887462 + 0.2333333334

Combine like terms: -0.2333333334 + 0.2333333334 = 0.0000000000
0.0000000000 + x = 0.276887462 + 0.2333333334
x = 0.276887462 + 0.2333333334

Combine like terms: 0.276887462 + 0.2333333334 = 0.5102207954
x = 0.5102207954

Simplifying
x = 0.5102207954

Subproblem 2
x + -0.2333333334 = -0.276887462

Simplifying
x + -0.2333333334 = -0.276887462

Reorder the terms:
-0.2333333334 + x = -0.276887462

Solving
-0.2333333334 + x = -0.276887462

Solving for variable 'x'.

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

Add '0.2333333334' to each side of the equation.
-0.2333333334 + 0.2333333334 + x = -0.276887462 + 0.2333333334

Combine like terms: -0.2333333334 + 0.2333333334 = 0.0000000000
0.0000000000 + x = -0.276887462 + 0.2333333334
x = -0.276887462 + 0.2333333334

Combine like terms: -0.276887462 + 0.2333333334 = -0.0435541286
x = -0.0435541286

Simplifying
x = -0.0435541286

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.5102207954, -0.0435541286}