32x^2+8x-1=0

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Solution for 32x^2+8x-1=0 equation:


Simplifying
32x2 + 8x + -1 = 0

Reorder the terms:
-1 + 8x + 32x2 = 0

Solving
-1 + 8x + 32x2 = 0

Solving for variable 'x'.

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

Divide each side by '32'.
-0.03125 + 0.25x + x2 = 0

Move the constant term to the right:

Add '0.03125' to each side of the equation.
-0.03125 + 0.25x + 0.03125 + x2 = 0 + 0.03125

Reorder the terms:
-0.03125 + 0.03125 + 0.25x + x2 = 0 + 0.03125

Combine like terms: -0.03125 + 0.03125 = 0.00000
0.00000 + 0.25x + x2 = 0 + 0.03125
0.25x + x2 = 0 + 0.03125

Combine like terms: 0 + 0.03125 = 0.03125
0.25x + x2 = 0.03125

The x term is 0.25x.  Take half its coefficient (0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
0.25x + 0.015625 + x2 = 0.03125 + 0.015625

Reorder the terms:
0.015625 + 0.25x + x2 = 0.03125 + 0.015625

Combine like terms: 0.03125 + 0.015625 = 0.046875
0.015625 + 0.25x + x2 = 0.046875

Factor a perfect square on the left side:
(x + 0.125)(x + 0.125) = 0.046875

Calculate the square root of the right side: 0.216506351

Break this problem into two subproblems by setting 
(x + 0.125) equal to 0.216506351 and -0.216506351.

Subproblem 1

x + 0.125 = 0.216506351 Simplifying x + 0.125 = 0.216506351 Reorder the terms: 0.125 + x = 0.216506351 Solving 0.125 + x = 0.216506351 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + x = 0.216506351 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + x = 0.216506351 + -0.125 x = 0.216506351 + -0.125 Combine like terms: 0.216506351 + -0.125 = 0.091506351 x = 0.091506351 Simplifying x = 0.091506351

Subproblem 2

x + 0.125 = -0.216506351 Simplifying x + 0.125 = -0.216506351 Reorder the terms: 0.125 + x = -0.216506351 Solving 0.125 + x = -0.216506351 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + x = -0.216506351 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + x = -0.216506351 + -0.125 x = -0.216506351 + -0.125 Combine like terms: -0.216506351 + -0.125 = -0.341506351 x = -0.341506351 Simplifying x = -0.341506351

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.091506351, -0.341506351}

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