2y^2-18y-23=0

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Solution for 2y^2-18y-23=0 equation:


Simplifying
2y2 + -18y + -23 = 0

Reorder the terms:
-23 + -18y + 2y2 = 0

Solving
-23 + -18y + 2y2 = 0

Solving for variable 'y'.

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

Divide each side by '2'.
-11.5 + -9y + y2 = 0

Move the constant term to the right:

Add '11.5' to each side of the equation.
-11.5 + -9y + 11.5 + y2 = 0 + 11.5

Reorder the terms:
-11.5 + 11.5 + -9y + y2 = 0 + 11.5

Combine like terms: -11.5 + 11.5 = 0.0
0.0 + -9y + y2 = 0 + 11.5
-9y + y2 = 0 + 11.5

Combine like terms: 0 + 11.5 = 11.5
-9y + y2 = 11.5

The y term is -9y.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9y + 20.25 + y2 = 11.5 + 20.25

Reorder the terms:
20.25 + -9y + y2 = 11.5 + 20.25

Combine like terms: 11.5 + 20.25 = 31.75
20.25 + -9y + y2 = 31.75

Factor a perfect square on the left side:
(y + -4.5)(y + -4.5) = 31.75

Calculate the square root of the right side: 5.634713835

Break this problem into two subproblems by setting 
(y + -4.5) equal to 5.634713835 and -5.634713835.

Subproblem 1

y + -4.5 = 5.634713835 Simplifying y + -4.5 = 5.634713835 Reorder the terms: -4.5 + y = 5.634713835 Solving -4.5 + y = 5.634713835 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + y = 5.634713835 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + y = 5.634713835 + 4.5 y = 5.634713835 + 4.5 Combine like terms: 5.634713835 + 4.5 = 10.134713835 y = 10.134713835 Simplifying y = 10.134713835

Subproblem 2

y + -4.5 = -5.634713835 Simplifying y + -4.5 = -5.634713835 Reorder the terms: -4.5 + y = -5.634713835 Solving -4.5 + y = -5.634713835 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + y = -5.634713835 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + y = -5.634713835 + 4.5 y = -5.634713835 + 4.5 Combine like terms: -5.634713835 + 4.5 = -1.134713835 y = -1.134713835 Simplifying y = -1.134713835

Solution

The solution to the problem is based on the solutions from the subproblems. y = {10.134713835, -1.134713835}

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