14x^2+77x+21=0

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Solution for 14x^2+77x+21=0 equation: 


Simplifying
14x2 + 77x + 21 = 0

Reorder the terms:
21 + 77x + 14x2 = 0

Solving
21 + 77x + 14x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '7'.
7(3 + 11x + 2x2) = 0

Ignore the factor 7.

Subproblem 1
Set the factor '(3 + 11x + 2x2)' equal to zero and attempt to solve:

Simplifying
3 + 11x + 2x2 = 0

Solving
3 + 11x + 2x2 = 0

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

Divide each side by '2'.
1.5 + 5.5x + x2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + 5.5x + -1.5 + x2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + 5.5x + x2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + 5.5x + x2 = 0 + -1.5
5.5x + x2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
5.5x + x2 = -1.5

The x term is 5.5x.  Take half its coefficient (2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
5.5x + 7.5625 + x2 = -1.5 + 7.5625

Reorder the terms:
7.5625 + 5.5x + x2 = -1.5 + 7.5625

Combine like terms: -1.5 + 7.5625 = 6.0625
7.5625 + 5.5x + x2 = 6.0625

Factor a perfect square on the left side:
(x + 2.75)(x + 2.75) = 6.0625

Calculate the square root of the right side: 2.46221445

Break this problem into two subproblems by setting 
(x + 2.75) equal to 2.46221445 and -2.46221445.


Subproblem 1
x + 2.75 = 2.46221445

Simplifying
x + 2.75 = 2.46221445

Reorder the terms:
2.75 + x = 2.46221445

Solving
2.75 + x = 2.46221445

Solving for variable 'x'.

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

Add '-2.75' to each side of the equation.
2.75 + -2.75 + x = 2.46221445 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + x = 2.46221445 + -2.75
x = 2.46221445 + -2.75

Combine like terms: 2.46221445 + -2.75 = -0.28778555
x = -0.28778555

Simplifying
x = -0.28778555


Subproblem 2
x + 2.75 = -2.46221445

Simplifying
x + 2.75 = -2.46221445

Reorder the terms:
2.75 + x = -2.46221445

Solving
2.75 + x = -2.46221445

Solving for variable 'x'.

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

Add '-2.75' to each side of the equation.
2.75 + -2.75 + x = -2.46221445 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + x = -2.46221445 + -2.75
x = -2.46221445 + -2.75

Combine like terms: -2.46221445 + -2.75 = -5.21221445
x = -5.21221445

Simplifying
x = -5.21221445


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.28778555, -5.21221445}
Solution
x = {-0.28778555, -5.21221445}

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