315y+3y^2=10

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Solution for 315y+3y^2=10 equation:


Simplifying
315y + 3y2 = 10

Solving
315y + 3y2 = 10

Solving for variable 'y'.

Reorder the terms:
-10 + 315y + 3y2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 315y + 3y2 = 0

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

Divide each side by '3'.
-3.333333333 + 105y + y2 = 0

Move the constant term to the right:

Add '3.333333333' to each side of the equation.
-3.333333333 + 105y + 3.333333333 + y2 = 0 + 3.333333333

Reorder the terms:
-3.333333333 + 3.333333333 + 105y + y2 = 0 + 3.333333333

Combine like terms: -3.333333333 + 3.333333333 = 0.000000000
0.000000000 + 105y + y2 = 0 + 3.333333333
105y + y2 = 0 + 3.333333333

Combine like terms: 0 + 3.333333333 = 3.333333333
105y + y2 = 3.333333333

The y term is 105y.  Take half its coefficient (52.5).
Square it (2756.25) and add it to both sides.

Add '2756.25' to each side of the equation.
105y + 2756.25 + y2 = 3.333333333 + 2756.25

Reorder the terms:
2756.25 + 105y + y2 = 3.333333333 + 2756.25

Combine like terms: 3.333333333 + 2756.25 = 2759.583333333
2756.25 + 105y + y2 = 2759.583333333

Factor a perfect square on the left side:
(y + 52.5)(y + 52.5) = 2759.583333333

Calculate the square root of the right side: 52.531736439

Break this problem into two subproblems by setting 
(y + 52.5) equal to 52.531736439 and -52.531736439.

Subproblem 1

y + 52.5 = 52.531736439 Simplifying y + 52.5 = 52.531736439 Reorder the terms: 52.5 + y = 52.531736439 Solving 52.5 + y = 52.531736439 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-52.5' to each side of the equation. 52.5 + -52.5 + y = 52.531736439 + -52.5 Combine like terms: 52.5 + -52.5 = 0.0 0.0 + y = 52.531736439 + -52.5 y = 52.531736439 + -52.5 Combine like terms: 52.531736439 + -52.5 = 0.031736439 y = 0.031736439 Simplifying y = 0.031736439

Subproblem 2

y + 52.5 = -52.531736439 Simplifying y + 52.5 = -52.531736439 Reorder the terms: 52.5 + y = -52.531736439 Solving 52.5 + y = -52.531736439 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-52.5' to each side of the equation. 52.5 + -52.5 + y = -52.531736439 + -52.5 Combine like terms: 52.5 + -52.5 = 0.0 0.0 + y = -52.531736439 + -52.5 y = -52.531736439 + -52.5 Combine like terms: -52.531736439 + -52.5 = -105.031736439 y = -105.031736439 Simplifying y = -105.031736439

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.031736439, -105.031736439}

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