(2x+3)2=(2x+1)(7-2x)+74

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Solution for (2x+3)2=(2x+1)(7-2x)+74 equation:


Simplifying
(2x + 3) * 2 = (2x + 1)(7 + -2x) + 74

Reorder the terms:
(3 + 2x) * 2 = (2x + 1)(7 + -2x) + 74

Reorder the terms for easier multiplication:
2(3 + 2x) = (2x + 1)(7 + -2x) + 74
(3 * 2 + 2x * 2) = (2x + 1)(7 + -2x) + 74
(6 + 4x) = (2x + 1)(7 + -2x) + 74

Reorder the terms:
6 + 4x = (1 + 2x)(7 + -2x) + 74

Multiply (1 + 2x) * (7 + -2x)
6 + 4x = (1(7 + -2x) + 2x * (7 + -2x)) + 74
6 + 4x = ((7 * 1 + -2x * 1) + 2x * (7 + -2x)) + 74
6 + 4x = ((7 + -2x) + 2x * (7 + -2x)) + 74
6 + 4x = (7 + -2x + (7 * 2x + -2x * 2x)) + 74
6 + 4x = (7 + -2x + (14x + -4x2)) + 74

Combine like terms: -2x + 14x = 12x
6 + 4x = (7 + 12x + -4x2) + 74

Reorder the terms:
6 + 4x = 7 + 74 + 12x + -4x2

Combine like terms: 7 + 74 = 81
6 + 4x = 81 + 12x + -4x2

Solving
6 + 4x = 81 + 12x + -4x2

Solving for variable 'x'.

Reorder the terms:
6 + -81 + 4x + -12x + 4x2 = 81 + 12x + -4x2 + -81 + -12x + 4x2

Combine like terms: 6 + -81 = -75
-75 + 4x + -12x + 4x2 = 81 + 12x + -4x2 + -81 + -12x + 4x2

Combine like terms: 4x + -12x = -8x
-75 + -8x + 4x2 = 81 + 12x + -4x2 + -81 + -12x + 4x2

Reorder the terms:
-75 + -8x + 4x2 = 81 + -81 + 12x + -12x + -4x2 + 4x2

Combine like terms: 81 + -81 = 0
-75 + -8x + 4x2 = 0 + 12x + -12x + -4x2 + 4x2
-75 + -8x + 4x2 = 12x + -12x + -4x2 + 4x2

Combine like terms: 12x + -12x = 0
-75 + -8x + 4x2 = 0 + -4x2 + 4x2
-75 + -8x + 4x2 = -4x2 + 4x2

Combine like terms: -4x2 + 4x2 = 0
-75 + -8x + 4x2 = 0

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

Divide each side by '4'.
-18.75 + -2x + x2 = 0

Move the constant term to the right:

Add '18.75' to each side of the equation.
-18.75 + -2x + 18.75 + x2 = 0 + 18.75

Reorder the terms:
-18.75 + 18.75 + -2x + x2 = 0 + 18.75

Combine like terms: -18.75 + 18.75 = 0.00
0.00 + -2x + x2 = 0 + 18.75
-2x + x2 = 0 + 18.75

Combine like terms: 0 + 18.75 = 18.75
-2x + x2 = 18.75

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 = 18.75 + 1

Reorder the terms:
1 + -2x + x2 = 18.75 + 1

Combine like terms: 18.75 + 1 = 19.75
1 + -2x + x2 = 19.75

Factor a perfect square on the left side:
(x + -1)(x + -1) = 19.75

Calculate the square root of the right side: 4.444097209

Break this problem into two subproblems by setting 
(x + -1) equal to 4.444097209 and -4.444097209.

Subproblem 1

x + -1 = 4.444097209 Simplifying x + -1 = 4.444097209 Reorder the terms: -1 + x = 4.444097209 Solving -1 + x = 4.444097209 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 = 4.444097209 + 1 Combine like terms: -1 + 1 = 0 0 + x = 4.444097209 + 1 x = 4.444097209 + 1 Combine like terms: 4.444097209 + 1 = 5.444097209 x = 5.444097209 Simplifying x = 5.444097209

Subproblem 2

x + -1 = -4.444097209 Simplifying x + -1 = -4.444097209 Reorder the terms: -1 + x = -4.444097209 Solving -1 + x = -4.444097209 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 = -4.444097209 + 1 Combine like terms: -1 + 1 = 0 0 + x = -4.444097209 + 1 x = -4.444097209 + 1 Combine like terms: -4.444097209 + 1 = -3.444097209 x = -3.444097209 Simplifying x = -3.444097209

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.444097209, -3.444097209}

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