6x+1=(4x+5)(2x-4)

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Solution for 6x+1=(4x+5)(2x-4) equation:


Simplifying
6x + 1 = (4x + 5)(2x + -4)

Reorder the terms:
1 + 6x = (4x + 5)(2x + -4)

Reorder the terms:
1 + 6x = (5 + 4x)(2x + -4)

Reorder the terms:
1 + 6x = (5 + 4x)(-4 + 2x)

Multiply (5 + 4x) * (-4 + 2x)
1 + 6x = (5(-4 + 2x) + 4x * (-4 + 2x))
1 + 6x = ((-4 * 5 + 2x * 5) + 4x * (-4 + 2x))
1 + 6x = ((-20 + 10x) + 4x * (-4 + 2x))
1 + 6x = (-20 + 10x + (-4 * 4x + 2x * 4x))
1 + 6x = (-20 + 10x + (-16x + 8x2))

Combine like terms: 10x + -16x = -6x
1 + 6x = (-20 + -6x + 8x2)

Solving
1 + 6x = -20 + -6x + 8x2

Solving for variable 'x'.

Reorder the terms:
1 + 20 + 6x + 6x + -8x2 = -20 + -6x + 8x2 + 20 + 6x + -8x2

Combine like terms: 1 + 20 = 21
21 + 6x + 6x + -8x2 = -20 + -6x + 8x2 + 20 + 6x + -8x2

Combine like terms: 6x + 6x = 12x
21 + 12x + -8x2 = -20 + -6x + 8x2 + 20 + 6x + -8x2

Reorder the terms:
21 + 12x + -8x2 = -20 + 20 + -6x + 6x + 8x2 + -8x2

Combine like terms: -20 + 20 = 0
21 + 12x + -8x2 = 0 + -6x + 6x + 8x2 + -8x2
21 + 12x + -8x2 = -6x + 6x + 8x2 + -8x2

Combine like terms: -6x + 6x = 0
21 + 12x + -8x2 = 0 + 8x2 + -8x2
21 + 12x + -8x2 = 8x2 + -8x2

Combine like terms: 8x2 + -8x2 = 0
21 + 12x + -8x2 = 0

Begin completing the square.  Divide all terms by
-8 the coefficient of the squared term: 

Divide each side by '-8'.
-2.625 + -1.5x + x2 = 0

Move the constant term to the right:

Add '2.625' to each side of the equation.
-2.625 + -1.5x + 2.625 + x2 = 0 + 2.625

Reorder the terms:
-2.625 + 2.625 + -1.5x + x2 = 0 + 2.625

Combine like terms: -2.625 + 2.625 = 0.000
0.000 + -1.5x + x2 = 0 + 2.625
-1.5x + x2 = 0 + 2.625

Combine like terms: 0 + 2.625 = 2.625
-1.5x + x2 = 2.625

The x term is -1.5x.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5x + 0.5625 + x2 = 2.625 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 2.625 + 0.5625

Combine like terms: 2.625 + 0.5625 = 3.1875
0.5625 + -1.5x + x2 = 3.1875

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 3.1875

Calculate the square root of the right side: 1.785357107

Break this problem into two subproblems by setting 
(x + -0.75) equal to 1.785357107 and -1.785357107.

Subproblem 1

x + -0.75 = 1.785357107 Simplifying x + -0.75 = 1.785357107 Reorder the terms: -0.75 + x = 1.785357107 Solving -0.75 + x = 1.785357107 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 1.785357107 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 1.785357107 + 0.75 x = 1.785357107 + 0.75 Combine like terms: 1.785357107 + 0.75 = 2.535357107 x = 2.535357107 Simplifying x = 2.535357107

Subproblem 2

x + -0.75 = -1.785357107 Simplifying x + -0.75 = -1.785357107 Reorder the terms: -0.75 + x = -1.785357107 Solving -0.75 + x = -1.785357107 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -1.785357107 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -1.785357107 + 0.75 x = -1.785357107 + 0.75 Combine like terms: -1.785357107 + 0.75 = -1.035357107 x = -1.035357107 Simplifying x = -1.035357107

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.535357107, -1.035357107}

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