(x-2)(4x+1)-(2x+3)2=8

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Solution for (x-2)(4x+1)-(2x+3)2=8 equation:


Simplifying
(x + -2)(4x + 1) + -1(2x + 3) * 2 = 8

Reorder the terms:
(-2 + x)(4x + 1) + -1(2x + 3) * 2 = 8

Reorder the terms:
(-2 + x)(1 + 4x) + -1(2x + 3) * 2 = 8

Multiply (-2 + x) * (1 + 4x)
(-2(1 + 4x) + x(1 + 4x)) + -1(2x + 3) * 2 = 8
((1 * -2 + 4x * -2) + x(1 + 4x)) + -1(2x + 3) * 2 = 8
((-2 + -8x) + x(1 + 4x)) + -1(2x + 3) * 2 = 8
(-2 + -8x + (1 * x + 4x * x)) + -1(2x + 3) * 2 = 8
(-2 + -8x + (1x + 4x2)) + -1(2x + 3) * 2 = 8

Combine like terms: -8x + 1x = -7x
(-2 + -7x + 4x2) + -1(2x + 3) * 2 = 8

Reorder the terms:
-2 + -7x + 4x2 + -1(3 + 2x) * 2 = 8

Reorder the terms for easier multiplication:
-2 + -7x + 4x2 + -1 * 2(3 + 2x) = 8

Multiply -1 * 2
-2 + -7x + 4x2 + -2(3 + 2x) = 8
-2 + -7x + 4x2 + (3 * -2 + 2x * -2) = 8
-2 + -7x + 4x2 + (-6 + -4x) = 8

Reorder the terms:
-2 + -6 + -7x + -4x + 4x2 = 8

Combine like terms: -2 + -6 = -8
-8 + -7x + -4x + 4x2 = 8

Combine like terms: -7x + -4x = -11x
-8 + -11x + 4x2 = 8

Solving
-8 + -11x + 4x2 = 8

Solving for variable 'x'.

Reorder the terms:
-8 + -8 + -11x + 4x2 = 8 + -8

Combine like terms: -8 + -8 = -16
-16 + -11x + 4x2 = 8 + -8

Combine like terms: 8 + -8 = 0
-16 + -11x + 4x2 = 0

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

Divide each side by '4'.
-4 + -2.75x + x2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -2.75x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + -2.75x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -2.75x + x2 = 0 + 4
-2.75x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
-2.75x + x2 = 4

The x term is -2.75x.  Take half its coefficient (-1.375).
Square it (1.890625) and add it to both sides.

Add '1.890625' to each side of the equation.
-2.75x + 1.890625 + x2 = 4 + 1.890625

Reorder the terms:
1.890625 + -2.75x + x2 = 4 + 1.890625

Combine like terms: 4 + 1.890625 = 5.890625
1.890625 + -2.75x + x2 = 5.890625

Factor a perfect square on the left side:
(x + -1.375)(x + -1.375) = 5.890625

Calculate the square root of the right side: 2.42706098

Break this problem into two subproblems by setting 
(x + -1.375) equal to 2.42706098 and -2.42706098.

Subproblem 1

x + -1.375 = 2.42706098 Simplifying x + -1.375 = 2.42706098 Reorder the terms: -1.375 + x = 2.42706098 Solving -1.375 + x = 2.42706098 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + x = 2.42706098 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + x = 2.42706098 + 1.375 x = 2.42706098 + 1.375 Combine like terms: 2.42706098 + 1.375 = 3.80206098 x = 3.80206098 Simplifying x = 3.80206098

Subproblem 2

x + -1.375 = -2.42706098 Simplifying x + -1.375 = -2.42706098 Reorder the terms: -1.375 + x = -2.42706098 Solving -1.375 + x = -2.42706098 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + x = -2.42706098 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + x = -2.42706098 + 1.375 x = -2.42706098 + 1.375 Combine like terms: -2.42706098 + 1.375 = -1.05206098 x = -1.05206098 Simplifying x = -1.05206098

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.80206098, -1.05206098}

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