16x(2-x)+(4x-5)2=1

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Solution for 16x(2-x)+(4x-5)2=1 equation:


Simplifying
16x(2 + -1x) + (4x + -5) * 2 = 1
(2 * 16x + -1x * 16x) + (4x + -5) * 2 = 1
(32x + -16x2) + (4x + -5) * 2 = 1

Reorder the terms:
32x + -16x2 + (-5 + 4x) * 2 = 1

Reorder the terms for easier multiplication:
32x + -16x2 + 2(-5 + 4x) = 1
32x + -16x2 + (-5 * 2 + 4x * 2) = 1
32x + -16x2 + (-10 + 8x) = 1

Reorder the terms:
-10 + 32x + 8x + -16x2 = 1

Combine like terms: 32x + 8x = 40x
-10 + 40x + -16x2 = 1

Solving
-10 + 40x + -16x2 = 1

Solving for variable 'x'.

Reorder the terms:
-10 + -1 + 40x + -16x2 = 1 + -1

Combine like terms: -10 + -1 = -11
-11 + 40x + -16x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-11 + 40x + -16x2 = 0

Begin completing the square.  Divide all terms by
-16 the coefficient of the squared term: 

Divide each side by '-16'.
0.6875 + -2.5x + x2 = 0

Move the constant term to the right:

Add '-0.6875' to each side of the equation.
0.6875 + -2.5x + -0.6875 + x2 = 0 + -0.6875

Reorder the terms:
0.6875 + -0.6875 + -2.5x + x2 = 0 + -0.6875

Combine like terms: 0.6875 + -0.6875 = 0.0000
0.0000 + -2.5x + x2 = 0 + -0.6875
-2.5x + x2 = 0 + -0.6875

Combine like terms: 0 + -0.6875 = -0.6875
-2.5x + x2 = -0.6875

The x term is -2.5x.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5x + 1.5625 + x2 = -0.6875 + 1.5625

Reorder the terms:
1.5625 + -2.5x + x2 = -0.6875 + 1.5625

Combine like terms: -0.6875 + 1.5625 = 0.875
1.5625 + -2.5x + x2 = 0.875

Factor a perfect square on the left side:
(x + -1.25)(x + -1.25) = 0.875

Calculate the square root of the right side: 0.935414347

Break this problem into two subproblems by setting 
(x + -1.25) equal to 0.935414347 and -0.935414347.

Subproblem 1

x + -1.25 = 0.935414347 Simplifying x + -1.25 = 0.935414347 Reorder the terms: -1.25 + x = 0.935414347 Solving -1.25 + x = 0.935414347 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 0.935414347 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 0.935414347 + 1.25 x = 0.935414347 + 1.25 Combine like terms: 0.935414347 + 1.25 = 2.185414347 x = 2.185414347 Simplifying x = 2.185414347

Subproblem 2

x + -1.25 = -0.935414347 Simplifying x + -1.25 = -0.935414347 Reorder the terms: -1.25 + x = -0.935414347 Solving -1.25 + x = -0.935414347 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -0.935414347 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -0.935414347 + 1.25 x = -0.935414347 + 1.25 Combine like terms: -0.935414347 + 1.25 = 0.314585653 x = 0.314585653 Simplifying x = 0.314585653

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.185414347, 0.314585653}

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