y^2+51y-338.4375=0

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Solution for y^2+51y-338.4375=0 equation: 


Simplifying
y2 + 51y + -338.4375 = 0

Reorder the terms:
-338.4375 + 51y + y2 = 0

Solving
-338.4375 + 51y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '338.4375' to each side of the equation.
-338.4375 + 51y + 338.4375 + y2 = 0 + 338.4375

Reorder the terms:
-338.4375 + 338.4375 + 51y + y2 = 0 + 338.4375

Combine like terms: -338.4375 + 338.4375 = 0.0000
0.0000 + 51y + y2 = 0 + 338.4375
51y + y2 = 0 + 338.4375

Combine like terms: 0 + 338.4375 = 338.4375
51y + y2 = 338.4375

The y term is 51y.  Take half its coefficient (25.5).
Square it (650.25) and add it to both sides.

Add '650.25' to each side of the equation.
51y + 650.25 + y2 = 338.4375 + 650.25

Reorder the terms:
650.25 + 51y + y2 = 338.4375 + 650.25

Combine like terms: 338.4375 + 650.25 = 988.6875
650.25 + 51y + y2 = 988.6875

Factor a perfect square on the left side:
(y + 25.5)(y + 25.5) = 988.6875

Calculate the square root of the right side: 31.443401534

Break this problem into two subproblems by setting 
(y + 25.5) equal to 31.443401534 and -31.443401534.


Subproblem 1
y + 25.5 = 31.443401534

Simplifying
y + 25.5 = 31.443401534

Reorder the terms:
25.5 + y = 31.443401534

Solving
25.5 + y = 31.443401534

Solving for variable 'y'.

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

Add '-25.5' to each side of the equation.
25.5 + -25.5 + y = 31.443401534 + -25.5

Combine like terms: 25.5 + -25.5 = 0.0
0.0 + y = 31.443401534 + -25.5
y = 31.443401534 + -25.5

Combine like terms: 31.443401534 + -25.5 = 5.943401534
y = 5.943401534

Simplifying
y = 5.943401534


Subproblem 2
y + 25.5 = -31.443401534

Simplifying
y + 25.5 = -31.443401534

Reorder the terms:
25.5 + y = -31.443401534

Solving
25.5 + y = -31.443401534

Solving for variable 'y'.

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

Add '-25.5' to each side of the equation.
25.5 + -25.5 + y = -31.443401534 + -25.5

Combine like terms: 25.5 + -25.5 = 0.0
0.0 + y = -31.443401534 + -25.5
y = -31.443401534 + -25.5

Combine like terms: -31.443401534 + -25.5 = -56.943401534
y = -56.943401534

Simplifying
y = -56.943401534


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {5.943401534, -56.943401534}

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