(2x+7)(2x-7)=2+4(x-5)2

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Solution for (2x+7)(2x-7)=2+4(x-5)2 equation:


Simplifying
(2x + 7)(2x + -7) = 2 + 4(x + -5) * 2

Reorder the terms:
(7 + 2x)(2x + -7) = 2 + 4(x + -5) * 2

Reorder the terms:
(7 + 2x)(-7 + 2x) = 2 + 4(x + -5) * 2

Multiply (7 + 2x) * (-7 + 2x)
(7(-7 + 2x) + 2x * (-7 + 2x)) = 2 + 4(x + -5) * 2
((-7 * 7 + 2x * 7) + 2x * (-7 + 2x)) = 2 + 4(x + -5) * 2
((-49 + 14x) + 2x * (-7 + 2x)) = 2 + 4(x + -5) * 2
(-49 + 14x + (-7 * 2x + 2x * 2x)) = 2 + 4(x + -5) * 2
(-49 + 14x + (-14x + 4x2)) = 2 + 4(x + -5) * 2

Combine like terms: 14x + -14x = 0
(-49 + 0 + 4x2) = 2 + 4(x + -5) * 2
(-49 + 4x2) = 2 + 4(x + -5) * 2

Reorder the terms:
-49 + 4x2 = 2 + 4(-5 + x) * 2

Reorder the terms for easier multiplication:
-49 + 4x2 = 2 + 4 * 2(-5 + x)

Multiply 4 * 2
-49 + 4x2 = 2 + 8(-5 + x)
-49 + 4x2 = 2 + (-5 * 8 + x * 8)
-49 + 4x2 = 2 + (-40 + 8x)

Combine like terms: 2 + -40 = -38
-49 + 4x2 = -38 + 8x

Solving
-49 + 4x2 = -38 + 8x

Solving for variable 'x'.

Reorder the terms:
-49 + 38 + -8x + 4x2 = -38 + 8x + 38 + -8x

Combine like terms: -49 + 38 = -11
-11 + -8x + 4x2 = -38 + 8x + 38 + -8x

Reorder the terms:
-11 + -8x + 4x2 = -38 + 38 + 8x + -8x

Combine like terms: -38 + 38 = 0
-11 + -8x + 4x2 = 0 + 8x + -8x
-11 + -8x + 4x2 = 8x + -8x

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

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

Divide each side by '4'.
-2.75 + -2x + x2 = 0

Move the constant term to the right:

Add '2.75' to each side of the equation.
-2.75 + -2x + 2.75 + x2 = 0 + 2.75

Reorder the terms:
-2.75 + 2.75 + -2x + x2 = 0 + 2.75

Combine like terms: -2.75 + 2.75 = 0.00
0.00 + -2x + x2 = 0 + 2.75
-2x + x2 = 0 + 2.75

Combine like terms: 0 + 2.75 = 2.75
-2x + x2 = 2.75

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 = 2.75 + 1

Reorder the terms:
1 + -2x + x2 = 2.75 + 1

Combine like terms: 2.75 + 1 = 3.75
1 + -2x + x2 = 3.75

Factor a perfect square on the left side:
(x + -1)(x + -1) = 3.75

Calculate the square root of the right side: 1.936491673

Break this problem into two subproblems by setting 
(x + -1) equal to 1.936491673 and -1.936491673.

Subproblem 1

x + -1 = 1.936491673 Simplifying x + -1 = 1.936491673 Reorder the terms: -1 + x = 1.936491673 Solving -1 + x = 1.936491673 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 = 1.936491673 + 1 Combine like terms: -1 + 1 = 0 0 + x = 1.936491673 + 1 x = 1.936491673 + 1 Combine like terms: 1.936491673 + 1 = 2.936491673 x = 2.936491673 Simplifying x = 2.936491673

Subproblem 2

x + -1 = -1.936491673 Simplifying x + -1 = -1.936491673 Reorder the terms: -1 + x = -1.936491673 Solving -1 + x = -1.936491673 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 = -1.936491673 + 1 Combine like terms: -1 + 1 = 0 0 + x = -1.936491673 + 1 x = -1.936491673 + 1 Combine like terms: -1.936491673 + 1 = -0.936491673 x = -0.936491673 Simplifying x = -0.936491673

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.936491673, -0.936491673}

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