14y^2=-19y+30

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Solution for 14y^2=-19y+30 equation:


Simplifying
14y2 = -19y + 30

Reorder the terms:
14y2 = 30 + -19y

Solving
14y2 = 30 + -19y

Solving for variable 'y'.

Reorder the terms:
-30 + 19y + 14y2 = 30 + -19y + -30 + 19y

Reorder the terms:
-30 + 19y + 14y2 = 30 + -30 + -19y + 19y

Combine like terms: 30 + -30 = 0
-30 + 19y + 14y2 = 0 + -19y + 19y
-30 + 19y + 14y2 = -19y + 19y

Combine like terms: -19y + 19y = 0
-30 + 19y + 14y2 = 0

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

Divide each side by '14'.
-2.142857143 + 1.357142857y + y2 = 0

Move the constant term to the right:

Add '2.142857143' to each side of the equation.
-2.142857143 + 1.357142857y + 2.142857143 + y2 = 0 + 2.142857143

Reorder the terms:
-2.142857143 + 2.142857143 + 1.357142857y + y2 = 0 + 2.142857143

Combine like terms: -2.142857143 + 2.142857143 = 0.000000000
0.000000000 + 1.357142857y + y2 = 0 + 2.142857143
1.357142857y + y2 = 0 + 2.142857143

Combine like terms: 0 + 2.142857143 = 2.142857143
1.357142857y + y2 = 2.142857143

The y term is 1.357142857y.  Take half its coefficient (0.6785714285).
Square it (0.4604591836) and add it to both sides.

Add '0.4604591836' to each side of the equation.
1.357142857y + 0.4604591836 + y2 = 2.142857143 + 0.4604591836

Reorder the terms:
0.4604591836 + 1.357142857y + y2 = 2.142857143 + 0.4604591836

Combine like terms: 2.142857143 + 0.4604591836 = 2.6033163266
0.4604591836 + 1.357142857y + y2 = 2.6033163266

Factor a perfect square on the left side:
(y + 0.6785714285)(y + 0.6785714285) = 2.6033163266

Calculate the square root of the right side: 1.613479571

Break this problem into two subproblems by setting 
(y + 0.6785714285) equal to 1.613479571 and -1.613479571.

Subproblem 1

y + 0.6785714285 = 1.613479571 Simplifying y + 0.6785714285 = 1.613479571 Reorder the terms: 0.6785714285 + y = 1.613479571 Solving 0.6785714285 + y = 1.613479571 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.6785714285' to each side of the equation. 0.6785714285 + -0.6785714285 + y = 1.613479571 + -0.6785714285 Combine like terms: 0.6785714285 + -0.6785714285 = 0.0000000000 0.0000000000 + y = 1.613479571 + -0.6785714285 y = 1.613479571 + -0.6785714285 Combine like terms: 1.613479571 + -0.6785714285 = 0.9349081425 y = 0.9349081425 Simplifying y = 0.9349081425

Subproblem 2

y + 0.6785714285 = -1.613479571 Simplifying y + 0.6785714285 = -1.613479571 Reorder the terms: 0.6785714285 + y = -1.613479571 Solving 0.6785714285 + y = -1.613479571 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.6785714285' to each side of the equation. 0.6785714285 + -0.6785714285 + y = -1.613479571 + -0.6785714285 Combine like terms: 0.6785714285 + -0.6785714285 = 0.0000000000 0.0000000000 + y = -1.613479571 + -0.6785714285 y = -1.613479571 + -0.6785714285 Combine like terms: -1.613479571 + -0.6785714285 = -2.2920509995 y = -2.2920509995 Simplifying y = -2.2920509995

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.9349081425, -2.2920509995}

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