y^2+4y-53=0

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Solution for y^2+4y-53=0 equation:


Simplifying
y2 + 4y + -53 = 0

Reorder the terms:
-53 + 4y + y2 = 0

Solving
-53 + 4y + 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 + 4y + 53 + y2 = 0 + 53

Reorder the terms:
-53 + 53 + 4y + y2 = 0 + 53

Combine like terms: -53 + 53 = 0
0 + 4y + y2 = 0 + 53
4y + y2 = 0 + 53

Combine like terms: 0 + 53 = 53
4y + y2 = 53

The y term is 4y.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4y + 4 + y2 = 53 + 4

Reorder the terms:
4 + 4y + y2 = 53 + 4

Combine like terms: 53 + 4 = 57
4 + 4y + y2 = 57

Factor a perfect square on the left side:
(y + 2)(y + 2) = 57

Calculate the square root of the right side: 7.549834435

Break this problem into two subproblems by setting 
(y + 2) equal to 7.549834435 and -7.549834435.

Subproblem 1

y + 2 = 7.549834435 Simplifying y + 2 = 7.549834435 Reorder the terms: 2 + y = 7.549834435 Solving 2 + y = 7.549834435 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + y = 7.549834435 + -2 Combine like terms: 2 + -2 = 0 0 + y = 7.549834435 + -2 y = 7.549834435 + -2 Combine like terms: 7.549834435 + -2 = 5.549834435 y = 5.549834435 Simplifying y = 5.549834435

Subproblem 2

y + 2 = -7.549834435 Simplifying y + 2 = -7.549834435 Reorder the terms: 2 + y = -7.549834435 Solving 2 + y = -7.549834435 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + y = -7.549834435 + -2 Combine like terms: 2 + -2 = 0 0 + y = -7.549834435 + -2 y = -7.549834435 + -2 Combine like terms: -7.549834435 + -2 = -9.549834435 y = -9.549834435 Simplifying y = -9.549834435

Solution

The solution to the problem is based on the solutions from the subproblems. y = {5.549834435, -9.549834435}

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