6x^2+84x-77=0

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Solution for 6x^2+84x-77=0 equation: 


Simplifying
6x2 + 84x + -77 = 0

Reorder the terms:
-77 + 84x + 6x2 = 0

Solving
-77 + 84x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-12.83333333 + 14x + x2 = 0

Move the constant term to the right:

Add '12.83333333' to each side of the equation.
-12.83333333 + 14x + 12.83333333 + x2 = 0 + 12.83333333

Reorder the terms:
-12.83333333 + 12.83333333 + 14x + x2 = 0 + 12.83333333

Combine like terms: -12.83333333 + 12.83333333 = 0.00000000
0.00000000 + 14x + x2 = 0 + 12.83333333
14x + x2 = 0 + 12.83333333

Combine like terms: 0 + 12.83333333 = 12.83333333
14x + x2 = 12.83333333

The x term is 14x.  Take half its coefficient (7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
14x + 49 + x2 = 12.83333333 + 49

Reorder the terms:
49 + 14x + x2 = 12.83333333 + 49

Combine like terms: 12.83333333 + 49 = 61.83333333
49 + 14x + x2 = 61.83333333

Factor a perfect square on the left side:
(x + 7)(x + 7) = 61.83333333

Calculate the square root of the right side: 7.863417408

Break this problem into two subproblems by setting 
(x + 7) equal to 7.863417408 and -7.863417408.


Subproblem 1
x + 7 = 7.863417408

Simplifying
x + 7 = 7.863417408

Reorder the terms:
7 + x = 7.863417408

Solving
7 + x = 7.863417408

Solving for variable 'x'.

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

Add '-7' to each side of the equation.
7 + -7 + x = 7.863417408 + -7

Combine like terms: 7 + -7 = 0
0 + x = 7.863417408 + -7
x = 7.863417408 + -7

Combine like terms: 7.863417408 + -7 = 0.863417408
x = 0.863417408

Simplifying
x = 0.863417408


Subproblem 2
x + 7 = -7.863417408

Simplifying
x + 7 = -7.863417408

Reorder the terms:
7 + x = -7.863417408

Solving
7 + x = -7.863417408

Solving for variable 'x'.

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

Add '-7' to each side of the equation.
7 + -7 + x = -7.863417408 + -7

Combine like terms: 7 + -7 = 0
0 + x = -7.863417408 + -7
x = -7.863417408 + -7

Combine like terms: -7.863417408 + -7 = -14.863417408
x = -14.863417408

Simplifying
x = -14.863417408


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.863417408, -14.863417408}

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