2x^2-77x-28=0

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Solution for 2x^2-77x-28=0 equation: 


Simplifying
2x2 + -77x + -28 = 0

Reorder the terms:
-28 + -77x + 2x2 = 0

Solving
-28 + -77x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
-14 + -38.5x + x2 = 0

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + -38.5x + 14 + x2 = 0 + 14

Reorder the terms:
-14 + 14 + -38.5x + x2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + -38.5x + x2 = 0 + 14
-38.5x + x2 = 0 + 14

Combine like terms: 0 + 14 = 14
-38.5x + x2 = 14

The x term is -38.5x.  Take half its coefficient (-19.25).
Square it (370.5625) and add it to both sides.

Add '370.5625' to each side of the equation.
-38.5x + 370.5625 + x2 = 14 + 370.5625

Reorder the terms:
370.5625 + -38.5x + x2 = 14 + 370.5625

Combine like terms: 14 + 370.5625 = 384.5625
370.5625 + -38.5x + x2 = 384.5625

Factor a perfect square on the left side:
(x + -19.25)(x + -19.25) = 384.5625

Calculate the square root of the right side: 19.610265169

Break this problem into two subproblems by setting 
(x + -19.25) equal to 19.610265169 and -19.610265169.


Subproblem 1
x + -19.25 = 19.610265169

Simplifying
x + -19.25 = 19.610265169

Reorder the terms:
-19.25 + x = 19.610265169

Solving
-19.25 + x = 19.610265169

Solving for variable 'x'.

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

Add '19.25' to each side of the equation.
-19.25 + 19.25 + x = 19.610265169 + 19.25

Combine like terms: -19.25 + 19.25 = 0.00
0.00 + x = 19.610265169 + 19.25
x = 19.610265169 + 19.25

Combine like terms: 19.610265169 + 19.25 = 38.860265169
x = 38.860265169

Simplifying
x = 38.860265169


Subproblem 2
x + -19.25 = -19.610265169

Simplifying
x + -19.25 = -19.610265169

Reorder the terms:
-19.25 + x = -19.610265169

Solving
-19.25 + x = -19.610265169

Solving for variable 'x'.

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

Add '19.25' to each side of the equation.
-19.25 + 19.25 + x = -19.610265169 + 19.25

Combine like terms: -19.25 + 19.25 = 0.00
0.00 + x = -19.610265169 + 19.25
x = -19.610265169 + 19.25

Combine like terms: -19.610265169 + 19.25 = -0.360265169
x = -0.360265169

Simplifying
x = -0.360265169


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {38.860265169, -0.360265169}

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