y^2-16y+52=0

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Solution for y^2-16y+52=0 equation:


Simplifying
y2 + -16y + 52 = 0

Reorder the terms:
52 + -16y + y2 = 0

Solving
52 + -16y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '-52' to each side of the equation.
52 + -16y + -52 + y2 = 0 + -52

Reorder the terms:
52 + -52 + -16y + y2 = 0 + -52

Combine like terms: 52 + -52 = 0
0 + -16y + y2 = 0 + -52
-16y + y2 = 0 + -52

Combine like terms: 0 + -52 = -52
-16y + y2 = -52

The y term is -16y.  Take half its coefficient (-8).
Square it (64) and add it to both sides.

Add '64' to each side of the equation.
-16y + 64 + y2 = -52 + 64

Reorder the terms:
64 + -16y + y2 = -52 + 64

Combine like terms: -52 + 64 = 12
64 + -16y + y2 = 12

Factor a perfect square on the left side:
(y + -8)(y + -8) = 12

Calculate the square root of the right side: 3.464101615

Break this problem into two subproblems by setting 
(y + -8) equal to 3.464101615 and -3.464101615.

Subproblem 1

y + -8 = 3.464101615 Simplifying y + -8 = 3.464101615 Reorder the terms: -8 + y = 3.464101615 Solving -8 + y = 3.464101615 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + y = 3.464101615 + 8 Combine like terms: -8 + 8 = 0 0 + y = 3.464101615 + 8 y = 3.464101615 + 8 Combine like terms: 3.464101615 + 8 = 11.464101615 y = 11.464101615 Simplifying y = 11.464101615

Subproblem 2

y + -8 = -3.464101615 Simplifying y + -8 = -3.464101615 Reorder the terms: -8 + y = -3.464101615 Solving -8 + y = -3.464101615 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + y = -3.464101615 + 8 Combine like terms: -8 + 8 = 0 0 + y = -3.464101615 + 8 y = -3.464101615 + 8 Combine like terms: -3.464101615 + 8 = 4.535898385 y = 4.535898385 Simplifying y = 4.535898385

Solution

The solution to the problem is based on the solutions from the subproblems. y = {11.464101615, 4.535898385}

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