(x+3)(x-4)52=-x

Solution for (x+3)(x-4)52=-x equation:

Simplifying
(x + 3)(x + -4) * 52 = -1x

Reorder the terms:
(3 + x)(x + -4) * 52 = -1x

Reorder the terms:
(3 + x)(-4 + x) * 52 = -1x

Reorder the terms for easier multiplication:
52(3 + x)(-4 + x) = -1x

Multiply (3 + x) * (-4 + x)
52(3(-4 + x) + x(-4 + x)) = -1x
52((-4 * 3 + x * 3) + x(-4 + x)) = -1x
52((-12 + 3x) + x(-4 + x)) = -1x
52(-12 + 3x + (-4 * x + x * x)) = -1x
52(-12 + 3x + (-4x + x2)) = -1x

Combine like terms: 3x + -4x = -1x
52(-12 + -1x + x2) = -1x
(-12 * 52 + -1x * 52 + x2 * 52) = -1x
(-624 + -52x + 52x2) = -1x

Solving
-624 + -52x + 52x2 = -1x

Solving for variable 'x'.

Reorder the terms:
-624 + -52x + x + 52x2 = -1x + x

Combine like terms: -52x + x = -51x
-624 + -51x + 52x2 = -1x + x

Combine like terms: -1x + x = 0
-624 + -51x + 52x2 = 0

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

Divide each side by '52'.
-12 + -0.9807692308x + x2 = 0

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + -0.9807692308x + 12 + x2 = 0 + 12

Reorder the terms:
-12 + 12 + -0.9807692308x + x2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + -0.9807692308x + x2 = 0 + 12
-0.9807692308x + x2 = 0 + 12

Combine like terms: 0 + 12 = 12
-0.9807692308x + x2 = 12

The x term is -0.9807692308x.  Take half its coefficient (-0.4903846154).
Square it (0.2404770710) and add it to both sides.

Add '0.2404770710' to each side of the equation.
-0.9807692308x + 0.2404770710 + x2 = 12 + 0.2404770710

Reorder the terms:
0.2404770710 + -0.9807692308x + x2 = 12 + 0.2404770710

Combine like terms: 12 + 0.2404770710 = 12.240477071
0.2404770710 + -0.9807692308x + x2 = 12.240477071

Factor a perfect square on the left side:
(x + -0.4903846154)(x + -0.4903846154) = 12.240477071

Calculate the square root of the right side: 3.498639317

Break this problem into two subproblems by setting 
(x + -0.4903846154) equal to 3.498639317 and -3.498639317.

Subproblem 1
x + -0.4903846154 = 3.498639317

Simplifying
x + -0.4903846154 = 3.498639317

Reorder the terms:
-0.4903846154 + x = 3.498639317

Solving
-0.4903846154 + x = 3.498639317

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '0.4903846154' to each side of the equation.
-0.4903846154 + 0.4903846154 + x = 3.498639317 + 0.4903846154

Combine like terms: -0.4903846154 + 0.4903846154 = 0.0000000000
0.0000000000 + x = 3.498639317 + 0.4903846154
x = 3.498639317 + 0.4903846154

Combine like terms: 3.498639317 + 0.4903846154 = 3.9890239324
x = 3.9890239324

Simplifying
x = 3.9890239324

Subproblem 2
x + -0.4903846154 = -3.498639317

Simplifying
x + -0.4903846154 = -3.498639317

Reorder the terms:
-0.4903846154 + x = -3.498639317

Solving
-0.4903846154 + x = -3.498639317

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '0.4903846154' to each side of the equation.
-0.4903846154 + 0.4903846154 + x = -3.498639317 + 0.4903846154

Combine like terms: -0.4903846154 + 0.4903846154 = 0.0000000000
0.0000000000 + x = -3.498639317 + 0.4903846154
x = -3.498639317 + 0.4903846154

Combine like terms: -3.498639317 + 0.4903846154 = -3.0082547016
x = -3.0082547016

Simplifying
x = -3.0082547016

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.9890239324, -3.0082547016}