-y^2+19y=1

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Solution for -y^2+19y=1 equation:


Simplifying
-1y2 + 19y = 1

Reorder the terms:
19y + -1y2 = 1

Solving
19y + -1y2 = 1

Solving for variable 'y'.

Reorder the terms:
-1 + 19y + -1y2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 19y + -1y2 = 0

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

Divide each side by '-1'.
1 + -19y + y2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -19y + -1 + y2 = 0 + -1

Reorder the terms:
1 + -1 + -19y + y2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -19y + y2 = 0 + -1
-19y + y2 = 0 + -1

Combine like terms: 0 + -1 = -1
-19y + y2 = -1

The y term is -19y.  Take half its coefficient (-9.5).
Square it (90.25) and add it to both sides.

Add '90.25' to each side of the equation.
-19y + 90.25 + y2 = -1 + 90.25

Reorder the terms:
90.25 + -19y + y2 = -1 + 90.25

Combine like terms: -1 + 90.25 = 89.25
90.25 + -19y + y2 = 89.25

Factor a perfect square on the left side:
(y + -9.5)(y + -9.5) = 89.25

Calculate the square root of the right side: 9.447221814

Break this problem into two subproblems by setting 
(y + -9.5) equal to 9.447221814 and -9.447221814.

Subproblem 1

y + -9.5 = 9.447221814 Simplifying y + -9.5 = 9.447221814 Reorder the terms: -9.5 + y = 9.447221814 Solving -9.5 + y = 9.447221814 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '9.5' to each side of the equation. -9.5 + 9.5 + y = 9.447221814 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + y = 9.447221814 + 9.5 y = 9.447221814 + 9.5 Combine like terms: 9.447221814 + 9.5 = 18.947221814 y = 18.947221814 Simplifying y = 18.947221814

Subproblem 2

y + -9.5 = -9.447221814 Simplifying y + -9.5 = -9.447221814 Reorder the terms: -9.5 + y = -9.447221814 Solving -9.5 + y = -9.447221814 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '9.5' to each side of the equation. -9.5 + 9.5 + y = -9.447221814 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + y = -9.447221814 + 9.5 y = -9.447221814 + 9.5 Combine like terms: -9.447221814 + 9.5 = 0.052778186 y = 0.052778186 Simplifying y = 0.052778186

Solution

The solution to the problem is based on the solutions from the subproblems. y = {18.947221814, 0.052778186}

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