(4x+1)(x-1)=1

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


Simplifying
(4x + 1)(x + -1) = 1

Reorder the terms:
(1 + 4x)(x + -1) = 1

Reorder the terms:
(1 + 4x)(-1 + x) = 1

Multiply (1 + 4x) * (-1 + x)
(1(-1 + x) + 4x * (-1 + x)) = 1
((-1 * 1 + x * 1) + 4x * (-1 + x)) = 1
((-1 + 1x) + 4x * (-1 + x)) = 1
(-1 + 1x + (-1 * 4x + x * 4x)) = 1
(-1 + 1x + (-4x + 4x2)) = 1

Combine like terms: 1x + -4x = -3x
(-1 + -3x + 4x2) = 1

Solving
-1 + -3x + 4x2 = 1

Solving for variable 'x'.

Reorder the terms:
-1 + -1 + -3x + 4x2 = 1 + -1

Combine like terms: -1 + -1 = -2
-2 + -3x + 4x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-2 + -3x + 4x2 = 0

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

Divide each side by '4'.
-0.5 + -0.75x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -0.75x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -0.75x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -0.75x + x2 = 0 + 0.5
-0.75x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-0.75x + x2 = 0.5

The x term is -0.75x.  Take half its coefficient (-0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
-0.75x + 0.140625 + x2 = 0.5 + 0.140625

Reorder the terms:
0.140625 + -0.75x + x2 = 0.5 + 0.140625

Combine like terms: 0.5 + 0.140625 = 0.640625
0.140625 + -0.75x + x2 = 0.640625

Factor a perfect square on the left side:
(x + -0.375)(x + -0.375) = 0.640625

Calculate the square root of the right side: 0.80039053

Break this problem into two subproblems by setting 
(x + -0.375) equal to 0.80039053 and -0.80039053.

Subproblem 1

x + -0.375 = 0.80039053 Simplifying x + -0.375 = 0.80039053 Reorder the terms: -0.375 + x = 0.80039053 Solving -0.375 + x = 0.80039053 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = 0.80039053 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = 0.80039053 + 0.375 x = 0.80039053 + 0.375 Combine like terms: 0.80039053 + 0.375 = 1.17539053 x = 1.17539053 Simplifying x = 1.17539053

Subproblem 2

x + -0.375 = -0.80039053 Simplifying x + -0.375 = -0.80039053 Reorder the terms: -0.375 + x = -0.80039053 Solving -0.375 + x = -0.80039053 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = -0.80039053 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = -0.80039053 + 0.375 x = -0.80039053 + 0.375 Combine like terms: -0.80039053 + 0.375 = -0.42539053 x = -0.42539053 Simplifying x = -0.42539053

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.17539053, -0.42539053}

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