0=y^2-18y-53

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Solution for 0=y^2-18y-53 equation:


Simplifying
0 = y2 + -18y + -53

Reorder the terms:
0 = -53 + -18y + y2

Solving
0 = -53 + -18y + y2

Solving for variable 'y'.

Combine like terms: 0 + 53 = 53
53 + 18y + -1y2 = -53 + -18y + y2 + 53 + 18y + -1y2

Reorder the terms:
53 + 18y + -1y2 = -53 + 53 + -18y + 18y + y2 + -1y2

Combine like terms: -53 + 53 = 0
53 + 18y + -1y2 = 0 + -18y + 18y + y2 + -1y2
53 + 18y + -1y2 = -18y + 18y + y2 + -1y2

Combine like terms: -18y + 18y = 0
53 + 18y + -1y2 = 0 + y2 + -1y2
53 + 18y + -1y2 = y2 + -1y2

Combine like terms: y2 + -1y2 = 0
53 + 18y + -1y2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-53 + -18y + y2 = 0

Move the constant term to the right:

Add '53' to each side of the equation.
-53 + -18y + 53 + y2 = 0 + 53

Reorder the terms:
-53 + 53 + -18y + y2 = 0 + 53

Combine like terms: -53 + 53 = 0
0 + -18y + y2 = 0 + 53
-18y + y2 = 0 + 53

Combine like terms: 0 + 53 = 53
-18y + y2 = 53

The y term is -18y.  Take half its coefficient (-9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
-18y + 81 + y2 = 53 + 81

Reorder the terms:
81 + -18y + y2 = 53 + 81

Combine like terms: 53 + 81 = 134
81 + -18y + y2 = 134

Factor a perfect square on the left side:
(y + -9)(y + -9) = 134

Calculate the square root of the right side: 11.575836903

Break this problem into two subproblems by setting 
(y + -9) equal to 11.575836903 and -11.575836903.

Subproblem 1

y + -9 = 11.575836903 Simplifying y + -9 = 11.575836903 Reorder the terms: -9 + y = 11.575836903 Solving -9 + y = 11.575836903 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + y = 11.575836903 + 9 Combine like terms: -9 + 9 = 0 0 + y = 11.575836903 + 9 y = 11.575836903 + 9 Combine like terms: 11.575836903 + 9 = 20.575836903 y = 20.575836903 Simplifying y = 20.575836903

Subproblem 2

y + -9 = -11.575836903 Simplifying y + -9 = -11.575836903 Reorder the terms: -9 + y = -11.575836903 Solving -9 + y = -11.575836903 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + y = -11.575836903 + 9 Combine like terms: -9 + 9 = 0 0 + y = -11.575836903 + 9 y = -11.575836903 + 9 Combine like terms: -11.575836903 + 9 = -2.575836903 y = -2.575836903 Simplifying y = -2.575836903

Solution

The solution to the problem is based on the solutions from the subproblems. y = {20.575836903, -2.575836903}

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