128x^2+80x-21=0

Solution for 128x^2+80x-21=0 equation:

Simplifying
128x2 + 80x + -21 = 0

Reorder the terms:
-21 + 80x + 128x2 = 0

Solving
-21 + 80x + 128x2 = 0

Solving for variable 'x'.

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

Divide each side by '128'.
-0.1640625 + 0.625x + x2 = 0

Move the constant term to the right:

Add '0.1640625' to each side of the equation.
-0.1640625 + 0.625x + 0.1640625 + x2 = 0 + 0.1640625

Reorder the terms:
-0.1640625 + 0.1640625 + 0.625x + x2 = 0 + 0.1640625

Combine like terms: -0.1640625 + 0.1640625 = 0.0000000
0.0000000 + 0.625x + x2 = 0 + 0.1640625
0.625x + x2 = 0 + 0.1640625

Combine like terms: 0 + 0.1640625 = 0.1640625
0.625x + x2 = 0.1640625

The x term is 0.625x.  Take half its coefficient (0.3125).
Square it (0.09765625) and add it to both sides.

Add '0.09765625' to each side of the equation.
0.625x + 0.09765625 + x2 = 0.1640625 + 0.09765625

Reorder the terms:
0.09765625 + 0.625x + x2 = 0.1640625 + 0.09765625

Combine like terms: 0.1640625 + 0.09765625 = 0.26171875
0.09765625 + 0.625x + x2 = 0.26171875

Factor a perfect square on the left side:
(x + 0.3125)(x + 0.3125) = 0.26171875

Calculate the square root of the right side: 0.511584548

Break this problem into two subproblems by setting 
(x + 0.3125) equal to 0.511584548 and -0.511584548.

Subproblem 1
x + 0.3125 = 0.511584548

Simplifying
x + 0.3125 = 0.511584548

Reorder the terms:
0.3125 + x = 0.511584548

Solving
0.3125 + x = 0.511584548

Solving for variable 'x'.

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

Add '-0.3125' to each side of the equation.
0.3125 + -0.3125 + x = 0.511584548 + -0.3125

Combine like terms: 0.3125 + -0.3125 = 0.0000
0.0000 + x = 0.511584548 + -0.3125
x = 0.511584548 + -0.3125

Combine like terms: 0.511584548 + -0.3125 = 0.199084548
x = 0.199084548

Simplifying
x = 0.199084548

Subproblem 2
x + 0.3125 = -0.511584548

Simplifying
x + 0.3125 = -0.511584548

Reorder the terms:
0.3125 + x = -0.511584548

Solving
0.3125 + x = -0.511584548

Solving for variable 'x'.

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

Add '-0.3125' to each side of the equation.
0.3125 + -0.3125 + x = -0.511584548 + -0.3125

Combine like terms: 0.3125 + -0.3125 = 0.0000
0.0000 + x = -0.511584548 + -0.3125
x = -0.511584548 + -0.3125

Combine like terms: -0.511584548 + -0.3125 = -0.824084548
x = -0.824084548

Simplifying
x = -0.824084548

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.199084548, -0.824084548}