2y+y^2=420

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Solution for 2y+y^2=420 equation:


Simplifying
2y + y2 = 420

Solving
2y + y2 = 420

Solving for variable 'y'.

Reorder the terms:
-420 + 2y + y2 = 420 + -420

Combine like terms: 420 + -420 = 0
-420 + 2y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '420' to each side of the equation.
-420 + 2y + 420 + y2 = 0 + 420

Reorder the terms:
-420 + 420 + 2y + y2 = 0 + 420

Combine like terms: -420 + 420 = 0
0 + 2y + y2 = 0 + 420
2y + y2 = 0 + 420

Combine like terms: 0 + 420 = 420
2y + y2 = 420

The y term is 2y.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2y + 1 + y2 = 420 + 1

Reorder the terms:
1 + 2y + y2 = 420 + 1

Combine like terms: 420 + 1 = 421
1 + 2y + y2 = 421

Factor a perfect square on the left side:
(y + 1)(y + 1) = 421

Calculate the square root of the right side: 20.518284529

Break this problem into two subproblems by setting 
(y + 1) equal to 20.518284529 and -20.518284529.

Subproblem 1

y + 1 = 20.518284529 Simplifying y + 1 = 20.518284529 Reorder the terms: 1 + y = 20.518284529 Solving 1 + y = 20.518284529 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = 20.518284529 + -1 Combine like terms: 1 + -1 = 0 0 + y = 20.518284529 + -1 y = 20.518284529 + -1 Combine like terms: 20.518284529 + -1 = 19.518284529 y = 19.518284529 Simplifying y = 19.518284529

Subproblem 2

y + 1 = -20.518284529 Simplifying y + 1 = -20.518284529 Reorder the terms: 1 + y = -20.518284529 Solving 1 + y = -20.518284529 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = -20.518284529 + -1 Combine like terms: 1 + -1 = 0 0 + y = -20.518284529 + -1 y = -20.518284529 + -1 Combine like terms: -20.518284529 + -1 = -21.518284529 y = -21.518284529 Simplifying y = -21.518284529

Solution

The solution to the problem is based on the solutions from the subproblems. y = {19.518284529, -21.518284529}

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