X^2+3X-37.7=0

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Solution for X^2+3X-37.7=0 equation: 


Simplifying
X2 + 3X + -37.7 = 0

Reorder the terms:
-37.7 + 3X + X2 = 0

Solving
-37.7 + 3X + X2 = 0

Solving for variable 'X'.

Begin completing the square.

Move the constant term to the right:

Add '37.7' to each side of the equation.
-37.7 + 3X + 37.7 + X2 = 0 + 37.7

Reorder the terms:
-37.7 + 37.7 + 3X + X2 = 0 + 37.7

Combine like terms: -37.7 + 37.7 = 0.0
0.0 + 3X + X2 = 0 + 37.7
3X + X2 = 0 + 37.7

Combine like terms: 0 + 37.7 = 37.7
3X + X2 = 37.7

The X term is 3X.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3X + 2.25 + X2 = 37.7 + 2.25

Reorder the terms:
2.25 + 3X + X2 = 37.7 + 2.25

Combine like terms: 37.7 + 2.25 = 39.95
2.25 + 3X + X2 = 39.95

Factor a perfect square on the left side:
(X + 1.5)(X + 1.5) = 39.95

Calculate the square root of the right side: 6.320601237

Break this problem into two subproblems by setting 
(X + 1.5) equal to 6.320601237 and -6.320601237.


Subproblem 1
X + 1.5 = 6.320601237

Simplifying
X + 1.5 = 6.320601237

Reorder the terms:
1.5 + X = 6.320601237

Solving
1.5 + X = 6.320601237

Solving for variable 'X'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + X = 6.320601237 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + X = 6.320601237 + -1.5
X = 6.320601237 + -1.5

Combine like terms: 6.320601237 + -1.5 = 4.820601237
X = 4.820601237

Simplifying
X = 4.820601237


Subproblem 2
X + 1.5 = -6.320601237

Simplifying
X + 1.5 = -6.320601237

Reorder the terms:
1.5 + X = -6.320601237

Solving
1.5 + X = -6.320601237

Solving for variable 'X'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + X = -6.320601237 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + X = -6.320601237 + -1.5
X = -6.320601237 + -1.5

Combine like terms: -6.320601237 + -1.5 = -7.820601237
X = -7.820601237

Simplifying
X = -7.820601237


Solution
The solution to the problem is based on the solutions
from the subproblems.
X = {4.820601237, -7.820601237}

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