y^2+64y=512

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Solution for y^2+64y=512 equation:


Simplifying
y2 + 64y = 512

Reorder the terms:
64y + y2 = 512

Solving
64y + y2 = 512

Solving for variable 'y'.

Reorder the terms:
-512 + 64y + y2 = 512 + -512

Combine like terms: 512 + -512 = 0
-512 + 64y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '512' to each side of the equation.
-512 + 64y + 512 + y2 = 0 + 512

Reorder the terms:
-512 + 512 + 64y + y2 = 0 + 512

Combine like terms: -512 + 512 = 0
0 + 64y + y2 = 0 + 512
64y + y2 = 0 + 512

Combine like terms: 0 + 512 = 512
64y + y2 = 512

The y term is 64y.  Take half its coefficient (32).
Square it (1024) and add it to both sides.

Add '1024' to each side of the equation.
64y + 1024 + y2 = 512 + 1024

Reorder the terms:
1024 + 64y + y2 = 512 + 1024

Combine like terms: 512 + 1024 = 1536
1024 + 64y + y2 = 1536

Factor a perfect square on the left side:
(y + 32)(y + 32) = 1536

Calculate the square root of the right side: 39.191835885

Break this problem into two subproblems by setting 
(y + 32) equal to 39.191835885 and -39.191835885.

Subproblem 1

y + 32 = 39.191835885 Simplifying y + 32 = 39.191835885 Reorder the terms: 32 + y = 39.191835885 Solving 32 + y = 39.191835885 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-32' to each side of the equation. 32 + -32 + y = 39.191835885 + -32 Combine like terms: 32 + -32 = 0 0 + y = 39.191835885 + -32 y = 39.191835885 + -32 Combine like terms: 39.191835885 + -32 = 7.191835885 y = 7.191835885 Simplifying y = 7.191835885

Subproblem 2

y + 32 = -39.191835885 Simplifying y + 32 = -39.191835885 Reorder the terms: 32 + y = -39.191835885 Solving 32 + y = -39.191835885 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-32' to each side of the equation. 32 + -32 + y = -39.191835885 + -32 Combine like terms: 32 + -32 = 0 0 + y = -39.191835885 + -32 y = -39.191835885 + -32 Combine like terms: -39.191835885 + -32 = -71.191835885 y = -71.191835885 Simplifying y = -71.191835885

Solution

The solution to the problem is based on the solutions from the subproblems. y = {7.191835885, -71.191835885}

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