8x-4x^2=1

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Solution for 8x-4x^2=1 equation:


Simplifying
8x + -4x2 = 1

Solving
8x + -4x2 = 1

Solving for variable 'x'.

Reorder the terms:
-1 + 8x + -4x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 8x + -4x2 = 0

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

Divide each side by '-4'.
0.25 + -2x + x2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + -2x + -0.25 + x2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + -2x + x2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + -2x + x2 = 0 + -0.25
-2x + x2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
-2x + x2 = -0.25

The x term is -2x.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x + 1 + x2 = -0.25 + 1

Reorder the terms:
1 + -2x + x2 = -0.25 + 1

Combine like terms: -0.25 + 1 = 0.75
1 + -2x + x2 = 0.75

Factor a perfect square on the left side:
(x + -1)(x + -1) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(x + -1) equal to 0.866025404 and -0.866025404.

Subproblem 1

x + -1 = 0.866025404 Simplifying x + -1 = 0.866025404 Reorder the terms: -1 + x = 0.866025404 Solving -1 + x = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 0.866025404 + 1 Combine like terms: -1 + 1 = 0 0 + x = 0.866025404 + 1 x = 0.866025404 + 1 Combine like terms: 0.866025404 + 1 = 1.866025404 x = 1.866025404 Simplifying x = 1.866025404

Subproblem 2

x + -1 = -0.866025404 Simplifying x + -1 = -0.866025404 Reorder the terms: -1 + x = -0.866025404 Solving -1 + x = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -0.866025404 + 1 Combine like terms: -1 + 1 = 0 0 + x = -0.866025404 + 1 x = -0.866025404 + 1 Combine like terms: -0.866025404 + 1 = 0.133974596 x = 0.133974596 Simplifying x = 0.133974596

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.866025404, 0.133974596}

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