4x^2-54x+77=0

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Solution for 4x^2-54x+77=0 equation: 


Simplifying
4x2 + -54x + 77 = 0

Reorder the terms:
77 + -54x + 4x2 = 0

Solving
77 + -54x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
19.25 + -13.5x + x2 = 0

Move the constant term to the right:

Add '-19.25' to each side of the equation.
19.25 + -13.5x + -19.25 + x2 = 0 + -19.25

Reorder the terms:
19.25 + -19.25 + -13.5x + x2 = 0 + -19.25

Combine like terms: 19.25 + -19.25 = 0.00
0.00 + -13.5x + x2 = 0 + -19.25
-13.5x + x2 = 0 + -19.25

Combine like terms: 0 + -19.25 = -19.25
-13.5x + x2 = -19.25

The x term is -13.5x.  Take half its coefficient (-6.75).
Square it (45.5625) and add it to both sides.

Add '45.5625' to each side of the equation.
-13.5x + 45.5625 + x2 = -19.25 + 45.5625

Reorder the terms:
45.5625 + -13.5x + x2 = -19.25 + 45.5625

Combine like terms: -19.25 + 45.5625 = 26.3125
45.5625 + -13.5x + x2 = 26.3125

Factor a perfect square on the left side:
(x + -6.75)(x + -6.75) = 26.3125

Calculate the square root of the right side: 5.129571132

Break this problem into two subproblems by setting 
(x + -6.75) equal to 5.129571132 and -5.129571132.


Subproblem 1
x + -6.75 = 5.129571132

Simplifying
x + -6.75 = 5.129571132

Reorder the terms:
-6.75 + x = 5.129571132

Solving
-6.75 + x = 5.129571132

Solving for variable 'x'.

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

Add '6.75' to each side of the equation.
-6.75 + 6.75 + x = 5.129571132 + 6.75

Combine like terms: -6.75 + 6.75 = 0.00
0.00 + x = 5.129571132 + 6.75
x = 5.129571132 + 6.75

Combine like terms: 5.129571132 + 6.75 = 11.879571132
x = 11.879571132

Simplifying
x = 11.879571132


Subproblem 2
x + -6.75 = -5.129571132

Simplifying
x + -6.75 = -5.129571132

Reorder the terms:
-6.75 + x = -5.129571132

Solving
-6.75 + x = -5.129571132

Solving for variable 'x'.

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

Add '6.75' to each side of the equation.
-6.75 + 6.75 + x = -5.129571132 + 6.75

Combine like terms: -6.75 + 6.75 = 0.00
0.00 + x = -5.129571132 + 6.75
x = -5.129571132 + 6.75

Combine like terms: -5.129571132 + 6.75 = 1.620428868
x = 1.620428868

Simplifying
x = 1.620428868


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {11.879571132, 1.620428868}

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