720x^2+648x-78=0

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Solution for 720x^2+648x-78=0 equation: 


Simplifying
720x2 + 648x + -78 = 0

Reorder the terms:
-78 + 648x + 720x2 = 0

Solving
-78 + 648x + 720x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '6'.
6(-13 + 108x + 120x2) = 0

Ignore the factor 6.

Subproblem 1
Set the factor '(-13 + 108x + 120x2)' equal to zero and attempt to solve:

Simplifying
-13 + 108x + 120x2 = 0

Solving
-13 + 108x + 120x2 = 0

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

Divide each side by '120'.
-0.1083333333 + 0.9x + x2 = 0.0

Move the constant term to the right:

Add '0.1083333333' to each side of the equation.
-0.1083333333 + 0.9x + 0.1083333333 + x2 = 0.0 + 0.1083333333

Reorder the terms:
-0.1083333333 + 0.1083333333 + 0.9x + x2 = 0.0 + 0.1083333333

Combine like terms: -0.1083333333 + 0.1083333333 = 0.0000000000
0.0000000000 + 0.9x + x2 = 0.0 + 0.1083333333
0.9x + x2 = 0.0 + 0.1083333333

Combine like terms: 0.0 + 0.1083333333 = 0.1083333333
0.9x + x2 = 0.1083333333

The x term is 0.9x.  Take half its coefficient (0.45).
Square it (0.2025) and add it to both sides.

Add '0.2025' to each side of the equation.
0.9x + 0.2025 + x2 = 0.1083333333 + 0.2025

Reorder the terms:
0.2025 + 0.9x + x2 = 0.1083333333 + 0.2025

Combine like terms: 0.1083333333 + 0.2025 = 0.3108333333
0.2025 + 0.9x + x2 = 0.3108333333

Factor a perfect square on the left side:
(x + 0.45)(x + 0.45) = 0.3108333333

Calculate the square root of the right side: 0.557524289

Break this problem into two subproblems by setting 
(x + 0.45) equal to 0.557524289 and -0.557524289.


Subproblem 1
x + 0.45 = 0.557524289

Simplifying
x + 0.45 = 0.557524289

Reorder the terms:
0.45 + x = 0.557524289

Solving
0.45 + x = 0.557524289

Solving for variable 'x'.

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

Add '-0.45' to each side of the equation.
0.45 + -0.45 + x = 0.557524289 + -0.45

Combine like terms: 0.45 + -0.45 = 0.00
0.00 + x = 0.557524289 + -0.45
x = 0.557524289 + -0.45

Combine like terms: 0.557524289 + -0.45 = 0.107524289
x = 0.107524289

Simplifying
x = 0.107524289


Subproblem 2
x + 0.45 = -0.557524289

Simplifying
x + 0.45 = -0.557524289

Reorder the terms:
0.45 + x = -0.557524289

Solving
0.45 + x = -0.557524289

Solving for variable 'x'.

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

Add '-0.45' to each side of the equation.
0.45 + -0.45 + x = -0.557524289 + -0.45

Combine like terms: 0.45 + -0.45 = 0.00
0.00 + x = -0.557524289 + -0.45
x = -0.557524289 + -0.45

Combine like terms: -0.557524289 + -0.45 = -1.007524289
x = -1.007524289

Simplifying
x = -1.007524289


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.107524289, -1.007524289}
Solution
x = {0.107524289, -1.007524289}

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