-32x^2-56x+416=0

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Solution for -32x^2-56x+416=0 equation: 


Simplifying
-32x2 + -56x + 416 = 0

Reorder the terms:
416 + -56x + -32x2 = 0

Solving
416 + -56x + -32x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '8'.
8(52 + -7x + -4x2) = 0

Ignore the factor 8.

Subproblem 1
Set the factor '(52 + -7x + -4x2)' equal to zero and attempt to solve:

Simplifying
52 + -7x + -4x2 = 0

Solving
52 + -7x + -4x2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-13 + 1.75x + x2 = 0

Move the constant term to the right:

Add '13' to each side of the equation.
-13 + 1.75x + 13 + x2 = 0 + 13

Reorder the terms:
-13 + 13 + 1.75x + x2 = 0 + 13

Combine like terms: -13 + 13 = 0
0 + 1.75x + x2 = 0 + 13
1.75x + x2 = 0 + 13

Combine like terms: 0 + 13 = 13
1.75x + x2 = 13

The x term is 1.75x.  Take half its coefficient (0.875).
Square it (0.765625) and add it to both sides.

Add '0.765625' to each side of the equation.
1.75x + 0.765625 + x2 = 13 + 0.765625

Reorder the terms:
0.765625 + 1.75x + x2 = 13 + 0.765625

Combine like terms: 13 + 0.765625 = 13.765625
0.765625 + 1.75x + x2 = 13.765625

Factor a perfect square on the left side:
(x + 0.875)(x + 0.875) = 13.765625

Calculate the square root of the right side: 3.71020552

Break this problem into two subproblems by setting 
(x + 0.875) equal to 3.71020552 and -3.71020552.


Subproblem 1
x + 0.875 = 3.71020552

Simplifying
x + 0.875 = 3.71020552

Reorder the terms:
0.875 + x = 3.71020552

Solving
0.875 + x = 3.71020552

Solving for variable 'x'.

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

Add '-0.875' to each side of the equation.
0.875 + -0.875 + x = 3.71020552 + -0.875

Combine like terms: 0.875 + -0.875 = 0.000
0.000 + x = 3.71020552 + -0.875
x = 3.71020552 + -0.875

Combine like terms: 3.71020552 + -0.875 = 2.83520552
x = 2.83520552

Simplifying
x = 2.83520552


Subproblem 2
x + 0.875 = -3.71020552

Simplifying
x + 0.875 = -3.71020552

Reorder the terms:
0.875 + x = -3.71020552

Solving
0.875 + x = -3.71020552

Solving for variable 'x'.

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

Add '-0.875' to each side of the equation.
0.875 + -0.875 + x = -3.71020552 + -0.875

Combine like terms: 0.875 + -0.875 = 0.000
0.000 + x = -3.71020552 + -0.875
x = -3.71020552 + -0.875

Combine like terms: -3.71020552 + -0.875 = -4.58520552
x = -4.58520552

Simplifying
x = -4.58520552


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.83520552, -4.58520552}
Solution
x = {2.83520552, -4.58520552}

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