10x-1=(2x*1)(2x+1)

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Solution for 10x-1=(2x*1)(2x+1) equation:


Simplifying
10x + -1 = (2x * 1)(2x + 1)

Reorder the terms:
-1 + 10x = (2x * 1)(2x + 1)

Reorder the terms for easier multiplication:
-1 + 10x = (2 * 1x)(2x + 1)

Multiply 2 * 1
-1 + 10x = (2x)(2x + 1)

Remove parenthesis around (2x)
-1 + 10x = 2x(2x + 1)

Reorder the terms:
-1 + 10x = 2x(1 + 2x)
-1 + 10x = (1 * 2x + 2x * 2x)
-1 + 10x = (2x + 4x2)

Solving
-1 + 10x = 2x + 4x2

Solving for variable 'x'.

Combine like terms: 10x + -2x = 8x
-1 + 8x + -4x2 = 2x + 4x2 + -2x + -4x2

Reorder the terms:
-1 + 8x + -4x2 = 2x + -2x + 4x2 + -4x2

Combine like terms: 2x + -2x = 0
-1 + 8x + -4x2 = 0 + 4x2 + -4x2
-1 + 8x + -4x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 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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