0=y^2-y-400

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Solution for 0=y^2-y-400 equation:


Simplifying
0 = y2 + -1y + -400

Reorder the terms:
0 = -400 + -1y + y2

Solving
0 = -400 + -1y + y2

Solving for variable 'y'.

Combine like terms: 0 + 400 = 400
400 + y + -1y2 = -400 + -1y + y2 + 400 + y + -1y2

Reorder the terms:
400 + y + -1y2 = -400 + 400 + -1y + y + y2 + -1y2

Combine like terms: -400 + 400 = 0
400 + y + -1y2 = 0 + -1y + y + y2 + -1y2
400 + y + -1y2 = -1y + y + y2 + -1y2

Combine like terms: -1y + y = 0
400 + y + -1y2 = 0 + y2 + -1y2
400 + y + -1y2 = y2 + -1y2

Combine like terms: y2 + -1y2 = 0
400 + y + -1y2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-400 + -1y + y2 = 0

Move the constant term to the right:

Add '400' to each side of the equation.
-400 + -1y + 400 + y2 = 0 + 400

Reorder the terms:
-400 + 400 + -1y + y2 = 0 + 400

Combine like terms: -400 + 400 = 0
0 + -1y + y2 = 0 + 400
-1y + y2 = 0 + 400

Combine like terms: 0 + 400 = 400
-1y + y2 = 400

The y term is y.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1y + 0.25 + y2 = 400 + 0.25

Reorder the terms:
0.25 + -1y + y2 = 400 + 0.25

Combine like terms: 400 + 0.25 = 400.25
0.25 + -1y + y2 = 400.25

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 400.25

Calculate the square root of the right side: 20.006249024

Break this problem into two subproblems by setting 
(y + 0.5) equal to 20.006249024 and -20.006249024.

Subproblem 1

y + 0.5 = 20.006249024 Simplifying y + 0.5 = 20.006249024 Reorder the terms: 0.5 + y = 20.006249024 Solving 0.5 + y = 20.006249024 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = 20.006249024 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 20.006249024 + -0.5 y = 20.006249024 + -0.5 Combine like terms: 20.006249024 + -0.5 = 19.506249024 y = 19.506249024 Simplifying y = 19.506249024

Subproblem 2

y + 0.5 = -20.006249024 Simplifying y + 0.5 = -20.006249024 Reorder the terms: 0.5 + y = -20.006249024 Solving 0.5 + y = -20.006249024 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = -20.006249024 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -20.006249024 + -0.5 y = -20.006249024 + -0.5 Combine like terms: -20.006249024 + -0.5 = -20.506249024 y = -20.506249024 Simplifying y = -20.506249024

Solution

The solution to the problem is based on the solutions from the subproblems. y = {19.506249024, -20.506249024}

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