y^2+2y-53=0

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Solution for y^2+2y-53=0 equation:


Simplifying
y2 + 2y + -53 = 0

Reorder the terms:
-53 + 2y + y2 = 0

Solving
-53 + 2y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '53' to each side of the equation.
-53 + 2y + 53 + y2 = 0 + 53

Reorder the terms:
-53 + 53 + 2y + y2 = 0 + 53

Combine like terms: -53 + 53 = 0
0 + 2y + y2 = 0 + 53
2y + y2 = 0 + 53

Combine like terms: 0 + 53 = 53
2y + y2 = 53

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 = 53 + 1

Reorder the terms:
1 + 2y + y2 = 53 + 1

Combine like terms: 53 + 1 = 54
1 + 2y + y2 = 54

Factor a perfect square on the left side:
(y + 1)(y + 1) = 54

Calculate the square root of the right side: 7.348469228

Break this problem into two subproblems by setting 
(y + 1) equal to 7.348469228 and -7.348469228.

Subproblem 1

y + 1 = 7.348469228 Simplifying y + 1 = 7.348469228 Reorder the terms: 1 + y = 7.348469228 Solving 1 + y = 7.348469228 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 = 7.348469228 + -1 Combine like terms: 1 + -1 = 0 0 + y = 7.348469228 + -1 y = 7.348469228 + -1 Combine like terms: 7.348469228 + -1 = 6.348469228 y = 6.348469228 Simplifying y = 6.348469228

Subproblem 2

y + 1 = -7.348469228 Simplifying y + 1 = -7.348469228 Reorder the terms: 1 + y = -7.348469228 Solving 1 + y = -7.348469228 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 = -7.348469228 + -1 Combine like terms: 1 + -1 = 0 0 + y = -7.348469228 + -1 y = -7.348469228 + -1 Combine like terms: -7.348469228 + -1 = -8.348469228 y = -8.348469228 Simplifying y = -8.348469228

Solution

The solution to the problem is based on the solutions from the subproblems. y = {6.348469228, -8.348469228}

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