-5x^2-40x-78=0

Solution for -5x^2-40x-78=0 equation:

Simplifying
-5x2 + -40x + -78 = 0

Reorder the terms:
-78 + -40x + -5x2 = 0

Solving
-78 + -40x + -5x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '-1'.
-1(78 + 40x + 5x2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(78 + 40x + 5x2)' equal to zero and attempt to solve:

Simplifying
78 + 40x + 5x2 = 0

Solving
78 + 40x + 5x2 = 0

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

Divide each side by '5'.
15.6 + 8x + x2 = 0

Move the constant term to the right:

Add '-15.6' to each side of the equation.
15.6 + 8x + -15.6 + x2 = 0 + -15.6

Reorder the terms:
15.6 + -15.6 + 8x + x2 = 0 + -15.6

Combine like terms: 15.6 + -15.6 = 0.0
0.0 + 8x + x2 = 0 + -15.6
8x + x2 = 0 + -15.6

Combine like terms: 0 + -15.6 = -15.6
8x + x2 = -15.6

The x term is 8x.  Take half its coefficient (4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
8x + 16 + x2 = -15.6 + 16

Reorder the terms:
16 + 8x + x2 = -15.6 + 16

Combine like terms: -15.6 + 16 = 0.4
16 + 8x + x2 = 0.4

Factor a perfect square on the left side:
(x + 4)(x + 4) = 0.4

Calculate the square root of the right side: 0.632455532

Break this problem into two subproblems by setting 
(x + 4) equal to 0.632455532 and -0.632455532.

Subproblem 1
x + 4 = 0.632455532

Simplifying
x + 4 = 0.632455532

Reorder the terms:
4 + x = 0.632455532

Solving
4 + x = 0.632455532

Solving for variable 'x'.

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

Add '-4' to each side of the equation.
4 + -4 + x = 0.632455532 + -4

Combine like terms: 4 + -4 = 0
0 + x = 0.632455532 + -4
x = 0.632455532 + -4

Combine like terms: 0.632455532 + -4 = -3.367544468
x = -3.367544468

Simplifying
x = -3.367544468

Subproblem 2
x + 4 = -0.632455532

Simplifying
x + 4 = -0.632455532

Reorder the terms:
4 + x = -0.632455532

Solving
4 + x = -0.632455532

Solving for variable 'x'.

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

Add '-4' to each side of the equation.
4 + -4 + x = -0.632455532 + -4

Combine like terms: 4 + -4 = 0
0 + x = -0.632455532 + -4
x = -0.632455532 + -4

Combine like terms: -0.632455532 + -4 = -4.632455532
x = -4.632455532

Simplifying
x = -4.632455532

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-3.367544468, -4.632455532}
Solution
x = {-3.367544468, -4.632455532}