4x^2+84x-774=0

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Solution for 4x^2+84x-774=0 equation: 


Simplifying
4x2 + 84x + -774 = 0

Reorder the terms:
-774 + 84x + 4x2 = 0

Solving
-774 + 84x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-387 + 42x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-387 + 42x + 2x2)' equal to zero and attempt to solve:

Simplifying
-387 + 42x + 2x2 = 0

Solving
-387 + 42x + 2x2 = 0

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

Divide each side by '2'.
-193.5 + 21x + x2 = 0

Move the constant term to the right:

Add '193.5' to each side of the equation.
-193.5 + 21x + 193.5 + x2 = 0 + 193.5

Reorder the terms:
-193.5 + 193.5 + 21x + x2 = 0 + 193.5

Combine like terms: -193.5 + 193.5 = 0.0
0.0 + 21x + x2 = 0 + 193.5
21x + x2 = 0 + 193.5

Combine like terms: 0 + 193.5 = 193.5
21x + x2 = 193.5

The x term is 21x.  Take half its coefficient (10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
21x + 110.25 + x2 = 193.5 + 110.25

Reorder the terms:
110.25 + 21x + x2 = 193.5 + 110.25

Combine like terms: 193.5 + 110.25 = 303.75
110.25 + 21x + x2 = 303.75

Factor a perfect square on the left side:
(x + 10.5)(x + 10.5) = 303.75

Calculate the square root of the right side: 17.428425058

Break this problem into two subproblems by setting 
(x + 10.5) equal to 17.428425058 and -17.428425058.


Subproblem 1
x + 10.5 = 17.428425058

Simplifying
x + 10.5 = 17.428425058

Reorder the terms:
10.5 + x = 17.428425058

Solving
10.5 + x = 17.428425058

Solving for variable 'x'.

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

Add '-10.5' to each side of the equation.
10.5 + -10.5 + x = 17.428425058 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + x = 17.428425058 + -10.5
x = 17.428425058 + -10.5

Combine like terms: 17.428425058 + -10.5 = 6.928425058
x = 6.928425058

Simplifying
x = 6.928425058


Subproblem 2
x + 10.5 = -17.428425058

Simplifying
x + 10.5 = -17.428425058

Reorder the terms:
10.5 + x = -17.428425058

Solving
10.5 + x = -17.428425058

Solving for variable 'x'.

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

Add '-10.5' to each side of the equation.
10.5 + -10.5 + x = -17.428425058 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + x = -17.428425058 + -10.5
x = -17.428425058 + -10.5

Combine like terms: -17.428425058 + -10.5 = -27.928425058
x = -27.928425058

Simplifying
x = -27.928425058


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.928425058, -27.928425058}
Solution
x = {6.928425058, -27.928425058}

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