(4x+5)(4x-5)=(4x-5)2+10

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Solution for (4x+5)(4x-5)=(4x-5)2+10 equation:


Simplifying
(4x + 5)(4x + -5) = (4x + -5) * 2 + 10

Reorder the terms:
(5 + 4x)(4x + -5) = (4x + -5) * 2 + 10

Reorder the terms:
(5 + 4x)(-5 + 4x) = (4x + -5) * 2 + 10

Multiply (5 + 4x) * (-5 + 4x)
(5(-5 + 4x) + 4x * (-5 + 4x)) = (4x + -5) * 2 + 10
((-5 * 5 + 4x * 5) + 4x * (-5 + 4x)) = (4x + -5) * 2 + 10
((-25 + 20x) + 4x * (-5 + 4x)) = (4x + -5) * 2 + 10
(-25 + 20x + (-5 * 4x + 4x * 4x)) = (4x + -5) * 2 + 10
(-25 + 20x + (-20x + 16x2)) = (4x + -5) * 2 + 10

Combine like terms: 20x + -20x = 0
(-25 + 0 + 16x2) = (4x + -5) * 2 + 10
(-25 + 16x2) = (4x + -5) * 2 + 10

Reorder the terms:
-25 + 16x2 = (-5 + 4x) * 2 + 10

Reorder the terms for easier multiplication:
-25 + 16x2 = 2(-5 + 4x) + 10
-25 + 16x2 = (-5 * 2 + 4x * 2) + 10
-25 + 16x2 = (-10 + 8x) + 10

Reorder the terms:
-25 + 16x2 = -10 + 10 + 8x

Combine like terms: -10 + 10 = 0
-25 + 16x2 = 0 + 8x
-25 + 16x2 = 8x

Solving
-25 + 16x2 = 8x

Solving for variable 'x'.

Reorder the terms:
-25 + -8x + 16x2 = 8x + -8x

Combine like terms: 8x + -8x = 0
-25 + -8x + 16x2 = 0

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

Divide each side by '16'.
-1.5625 + -0.5x + x2 = 0

Move the constant term to the right:

Add '1.5625' to each side of the equation.
-1.5625 + -0.5x + 1.5625 + x2 = 0 + 1.5625

Reorder the terms:
-1.5625 + 1.5625 + -0.5x + x2 = 0 + 1.5625

Combine like terms: -1.5625 + 1.5625 = 0.0000
0.0000 + -0.5x + x2 = 0 + 1.5625
-0.5x + x2 = 0 + 1.5625

Combine like terms: 0 + 1.5625 = 1.5625
-0.5x + x2 = 1.5625

The x term is -0.5x.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x + 0.0625 + x2 = 1.5625 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 1.5625 + 0.0625

Combine like terms: 1.5625 + 0.0625 = 1.625
0.0625 + -0.5x + x2 = 1.625

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 1.625

Calculate the square root of the right side: 1.274754878

Break this problem into two subproblems by setting 
(x + -0.25) equal to 1.274754878 and -1.274754878.

Subproblem 1

x + -0.25 = 1.274754878 Simplifying x + -0.25 = 1.274754878 Reorder the terms: -0.25 + x = 1.274754878 Solving -0.25 + x = 1.274754878 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 1.274754878 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 1.274754878 + 0.25 x = 1.274754878 + 0.25 Combine like terms: 1.274754878 + 0.25 = 1.524754878 x = 1.524754878 Simplifying x = 1.524754878

Subproblem 2

x + -0.25 = -1.274754878 Simplifying x + -0.25 = -1.274754878 Reorder the terms: -0.25 + x = -1.274754878 Solving -0.25 + x = -1.274754878 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -1.274754878 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -1.274754878 + 0.25 x = -1.274754878 + 0.25 Combine like terms: -1.274754878 + 0.25 = -1.024754878 x = -1.024754878 Simplifying x = -1.024754878

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.524754878, -1.024754878}

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