12x^2-77x+20=0

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Solution for 12x^2-77x+20=0 equation:


Simplifying
12x2 + -77x + 20 = 0

Reorder the terms:
20 + -77x + 12x2 = 0

Solving
20 + -77x + 12x2 = 0

Solving for variable 'x'.

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

Divide each side by '12'.
1.666666667 + -6.416666667x + x2 = 0

Move the constant term to the right:

Add '-1.666666667' to each side of the equation.
1.666666667 + -6.416666667x + -1.666666667 + x2 = 0 + -1.666666667

Reorder the terms:
1.666666667 + -1.666666667 + -6.416666667x + x2 = 0 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + -6.416666667x + x2 = 0 + -1.666666667
-6.416666667x + x2 = 0 + -1.666666667

Combine like terms: 0 + -1.666666667 = -1.666666667
-6.416666667x + x2 = -1.666666667

The x term is -6.416666667x.  Take half its coefficient (-3.208333334).
Square it (10.29340278) and add it to both sides.

Add '10.29340278' to each side of the equation.
-6.416666667x + 10.29340278 + x2 = -1.666666667 + 10.29340278

Reorder the terms:
10.29340278 + -6.416666667x + x2 = -1.666666667 + 10.29340278

Combine like terms: -1.666666667 + 10.29340278 = 8.626736113
10.29340278 + -6.416666667x + x2 = 8.626736113

Factor a perfect square on the left side:
(x + -3.208333334)(x + -3.208333334) = 8.626736113

Calculate the square root of the right side: 2.937130592

Break this problem into two subproblems by setting 
(x + -3.208333334) equal to 2.937130592 and -2.937130592.


Subproblem 1
x + -3.208333334 = 2.937130592

Simplifying
x + -3.208333334 = 2.937130592

Reorder the terms:
-3.208333334 + x = 2.937130592

Solving
-3.208333334 + x = 2.937130592

Solving for variable 'x'.

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

Add '3.208333334' to each side of the equation.
-3.208333334 + 3.208333334 + x = 2.937130592 + 3.208333334

Combine like terms: -3.208333334 + 3.208333334 = 0.000000000
0.000000000 + x = 2.937130592 + 3.208333334
x = 2.937130592 + 3.208333334

Combine like terms: 2.937130592 + 3.208333334 = 6.145463926
x = 6.145463926

Simplifying
x = 6.145463926


Subproblem 2
x + -3.208333334 = -2.937130592

Simplifying
x + -3.208333334 = -2.937130592

Reorder the terms:
-3.208333334 + x = -2.937130592

Solving
-3.208333334 + x = -2.937130592

Solving for variable 'x'.

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

Add '3.208333334' to each side of the equation.
-3.208333334 + 3.208333334 + x = -2.937130592 + 3.208333334

Combine like terms: -3.208333334 + 3.208333334 = 0.000000000
0.000000000 + x = -2.937130592 + 3.208333334
x = -2.937130592 + 3.208333334

Combine like terms: -2.937130592 + 3.208333334 = 0.271202742
x = 0.271202742

Simplifying
x = 0.271202742


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.145463926, 0.271202742}

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