y-19y^2=-60

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Solution for y-19y^2=-60 equation:


Simplifying
y + -19y2 = -60

Solving
y + -19y2 = -60

Solving for variable 'y'.

Reorder the terms:
60 + y + -19y2 = -60 + 60

Combine like terms: -60 + 60 = 0
60 + y + -19y2 = 0

Begin completing the square.  Divide all terms by
-19 the coefficient of the squared term: 

Divide each side by '-19'.
-3.157894737 + -0.05263157895y + y2 = 0

Move the constant term to the right:

Add '3.157894737' to each side of the equation.
-3.157894737 + -0.05263157895y + 3.157894737 + y2 = 0 + 3.157894737

Reorder the terms:
-3.157894737 + 3.157894737 + -0.05263157895y + y2 = 0 + 3.157894737

Combine like terms: -3.157894737 + 3.157894737 = 0.000000000
0.000000000 + -0.05263157895y + y2 = 0 + 3.157894737
-0.05263157895y + y2 = 0 + 3.157894737

Combine like terms: 0 + 3.157894737 = 3.157894737
-0.05263157895y + y2 = 3.157894737

The y term is y.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-0.05263157895y + 0.25 + y2 = 3.157894737 + 0.25

Reorder the terms:
0.25 + -0.05263157895y + y2 = 3.157894737 + 0.25

Combine like terms: 3.157894737 + 0.25 = 3.407894737
0.25 + -0.05263157895y + y2 = 3.407894737

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 3.407894737

Calculate the square root of the right side: 1.846048411

Break this problem into two subproblems by setting 
(y + 0.5) equal to 1.846048411 and -1.846048411.

Subproblem 1

y + 0.5 = 1.846048411 Simplifying y + 0.5 = 1.846048411 Reorder the terms: 0.5 + y = 1.846048411 Solving 0.5 + y = 1.846048411 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = 1.846048411 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 1.846048411 + -0.5 y = 1.846048411 + -0.5 Combine like terms: 1.846048411 + -0.5 = 1.346048411 y = 1.346048411 Simplifying y = 1.346048411

Subproblem 2

y + 0.5 = -1.846048411 Simplifying y + 0.5 = -1.846048411 Reorder the terms: 0.5 + y = -1.846048411 Solving 0.5 + y = -1.846048411 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = -1.846048411 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -1.846048411 + -0.5 y = -1.846048411 + -0.5 Combine like terms: -1.846048411 + -0.5 = -2.346048411 y = -2.346048411 Simplifying y = -2.346048411

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.346048411, -2.346048411}

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