4*(x+1)*x=1024

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Solution for 4*(x+1)*x=1024 equation: 


Simplifying
4(x + 1) * x = 1024

Reorder the terms:
4(1 + x) * x = 1024

Reorder the terms for easier multiplication:
4x(1 + x) = 1024
(1 * 4x + x * 4x) = 1024
(4x + 4x2) = 1024

Solving
4x + 4x2 = 1024

Solving for variable 'x'.

Reorder the terms:
-1024 + 4x + 4x2 = 1024 + -1024

Combine like terms: 1024 + -1024 = 0
-1024 + 4x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-256 + x + x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-256 + x + x2)' equal to zero and attempt to solve:

Simplifying
-256 + x + x2 = 0

Solving
-256 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '256' to each side of the equation.
-256 + x + 256 + x2 = 0 + 256

Reorder the terms:
-256 + 256 + x + x2 = 0 + 256

Combine like terms: -256 + 256 = 0
0 + x + x2 = 0 + 256
x + x2 = 0 + 256

Combine like terms: 0 + 256 = 256
x + x2 = 256

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
x + 0.25 + x2 = 256 + 0.25

Reorder the terms:
0.25 + x + x2 = 256 + 0.25

Combine like terms: 256 + 0.25 = 256.25
0.25 + x + x2 = 256.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 256.25

Calculate the square root of the right side: 16.007810594

Break this problem into two subproblems by setting 
(x + 0.5) equal to 16.007810594 and -16.007810594.


Subproblem 1
x + 0.5 = 16.007810594

Simplifying
x + 0.5 = 16.007810594

Reorder the terms:
0.5 + x = 16.007810594

Solving
0.5 + x = 16.007810594

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 16.007810594 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 16.007810594 + -0.5
x = 16.007810594 + -0.5

Combine like terms: 16.007810594 + -0.5 = 15.507810594
x = 15.507810594

Simplifying
x = 15.507810594


Subproblem 2
x + 0.5 = -16.007810594

Simplifying
x + 0.5 = -16.007810594

Reorder the terms:
0.5 + x = -16.007810594

Solving
0.5 + x = -16.007810594

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -16.007810594 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -16.007810594 + -0.5
x = -16.007810594 + -0.5

Combine like terms: -16.007810594 + -0.5 = -16.507810594
x = -16.507810594

Simplifying
x = -16.507810594


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {15.507810594, -16.507810594}
Solution
x = {15.507810594, -16.507810594}

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