-12X^2+72X+78=0

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Solution for -12X^2+72X+78=0 equation: 


Simplifying
-12X2 + 72X + 78 = 0

Reorder the terms:
78 + 72X + -12X2 = 0

Solving
78 + 72X + -12X2 = 0

Solving for variable 'X'.

Factor out the Greatest Common Factor (GCF), '6'.
6(13 + 12X + -2X2) = 0

Ignore the factor 6.

Subproblem 1
Set the factor '(13 + 12X + -2X2)' equal to zero and attempt to solve:

Simplifying
13 + 12X + -2X2 = 0

Solving
13 + 12X + -2X2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-6.5 + -6X + X2 = 0

Move the constant term to the right:

Add '6.5' to each side of the equation.
-6.5 + -6X + 6.5 + X2 = 0 + 6.5

Reorder the terms:
-6.5 + 6.5 + -6X + X2 = 0 + 6.5

Combine like terms: -6.5 + 6.5 = 0.0
0.0 + -6X + X2 = 0 + 6.5
-6X + X2 = 0 + 6.5

Combine like terms: 0 + 6.5 = 6.5
-6X + X2 = 6.5

The X term is -6X.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6X + 9 + X2 = 6.5 + 9

Reorder the terms:
9 + -6X + X2 = 6.5 + 9

Combine like terms: 6.5 + 9 = 15.5
9 + -6X + X2 = 15.5

Factor a perfect square on the left side:
(X + -3)(X + -3) = 15.5

Calculate the square root of the right side: 3.937003937

Break this problem into two subproblems by setting 
(X + -3) equal to 3.937003937 and -3.937003937.


Subproblem 1
X + -3 = 3.937003937

Simplifying
X + -3 = 3.937003937

Reorder the terms:
-3 + X = 3.937003937

Solving
-3 + X = 3.937003937

Solving for variable 'X'.

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

Add '3' to each side of the equation.
-3 + 3 + X = 3.937003937 + 3

Combine like terms: -3 + 3 = 0
0 + X = 3.937003937 + 3
X = 3.937003937 + 3

Combine like terms: 3.937003937 + 3 = 6.937003937
X = 6.937003937

Simplifying
X = 6.937003937


Subproblem 2
X + -3 = -3.937003937

Simplifying
X + -3 = -3.937003937

Reorder the terms:
-3 + X = -3.937003937

Solving
-3 + X = -3.937003937

Solving for variable 'X'.

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

Add '3' to each side of the equation.
-3 + 3 + X = -3.937003937 + 3

Combine like terms: -3 + 3 = 0
0 + X = -3.937003937 + 3
X = -3.937003937 + 3

Combine like terms: -3.937003937 + 3 = -0.937003937
X = -0.937003937

Simplifying
X = -0.937003937


Solution
The solution to the problem is based on the solutions
from the subproblems.
X = {6.937003937, -0.937003937}
Solution
X = {6.937003937, -0.937003937}

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