37x^2-21x-11.25=0

Solution for 37x^2-21x-11.25=0 equation:

Simplifying
37x2 + -21x + -11.25 = 0

Reorder the terms:
-11.25 + -21x + 37x2 = 0

Solving
-11.25 + -21x + 37x2 = 0

Solving for variable 'x'.

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

Divide each side by '37'.
-0.3040540541 + -0.5675675676x + x2 = 0

Move the constant term to the right:

Add '0.3040540541' to each side of the equation.
-0.3040540541 + -0.5675675676x + 0.3040540541 + x2 = 0 + 0.3040540541

Reorder the terms:
-0.3040540541 + 0.3040540541 + -0.5675675676x + x2 = 0 + 0.3040540541

Combine like terms: -0.3040540541 + 0.3040540541 = 0.0000000000
0.0000000000 + -0.5675675676x + x2 = 0 + 0.3040540541
-0.5675675676x + x2 = 0 + 0.3040540541

Combine like terms: 0 + 0.3040540541 = 0.3040540541
-0.5675675676x + x2 = 0.3040540541

The x term is -0.5675675676x.  Take half its coefficient (-0.2837837838).
Square it (0.08053323595) and add it to both sides.

Add '0.08053323595' to each side of the equation.
-0.5675675676x + 0.08053323595 + x2 = 0.3040540541 + 0.08053323595

Reorder the terms:
0.08053323595 + -0.5675675676x + x2 = 0.3040540541 + 0.08053323595

Combine like terms: 0.3040540541 + 0.08053323595 = 0.38458729005
0.08053323595 + -0.5675675676x + x2 = 0.38458729005

Factor a perfect square on the left side:
(x + -0.2837837838)(x + -0.2837837838) = 0.38458729005

Calculate the square root of the right side: 0.620151022

Break this problem into two subproblems by setting 
(x + -0.2837837838) equal to 0.620151022 and -0.620151022.

Subproblem 1
x + -0.2837837838 = 0.620151022

Simplifying
x + -0.2837837838 = 0.620151022

Reorder the terms:
-0.2837837838 + x = 0.620151022

Solving
-0.2837837838 + x = 0.620151022

Solving for variable 'x'.

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

Add '0.2837837838' to each side of the equation.
-0.2837837838 + 0.2837837838 + x = 0.620151022 + 0.2837837838

Combine like terms: -0.2837837838 + 0.2837837838 = 0.0000000000
0.0000000000 + x = 0.620151022 + 0.2837837838
x = 0.620151022 + 0.2837837838

Combine like terms: 0.620151022 + 0.2837837838 = 0.9039348058
x = 0.9039348058

Simplifying
x = 0.9039348058

Subproblem 2
x + -0.2837837838 = -0.620151022

Simplifying
x + -0.2837837838 = -0.620151022

Reorder the terms:
-0.2837837838 + x = -0.620151022

Solving
-0.2837837838 + x = -0.620151022

Solving for variable 'x'.

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

Add '0.2837837838' to each side of the equation.
-0.2837837838 + 0.2837837838 + x = -0.620151022 + 0.2837837838

Combine like terms: -0.2837837838 + 0.2837837838 = 0.0000000000
0.0000000000 + x = -0.620151022 + 0.2837837838
x = -0.620151022 + 0.2837837838

Combine like terms: -0.620151022 + 0.2837837838 = -0.3363672382
x = -0.3363672382

Simplifying
x = -0.3363672382

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.9039348058, -0.3363672382}