10y^2=16y+5

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Solution for 10y^2=16y+5 equation:


Simplifying
10y2 = 16y + 5

Reorder the terms:
10y2 = 5 + 16y

Solving
10y2 = 5 + 16y

Solving for variable 'y'.

Reorder the terms:
-5 + -16y + 10y2 = 5 + 16y + -5 + -16y

Reorder the terms:
-5 + -16y + 10y2 = 5 + -5 + 16y + -16y

Combine like terms: 5 + -5 = 0
-5 + -16y + 10y2 = 0 + 16y + -16y
-5 + -16y + 10y2 = 16y + -16y

Combine like terms: 16y + -16y = 0
-5 + -16y + 10y2 = 0

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

Divide each side by '10'.
-0.5 + -1.6y + y2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -1.6y + 0.5 + y2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -1.6y + y2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -1.6y + y2 = 0 + 0.5
-1.6y + y2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-1.6y + y2 = 0.5

The y term is -1.6y.  Take half its coefficient (-0.8).
Square it (0.64) and add it to both sides.

Add '0.64' to each side of the equation.
-1.6y + 0.64 + y2 = 0.5 + 0.64

Reorder the terms:
0.64 + -1.6y + y2 = 0.5 + 0.64

Combine like terms: 0.5 + 0.64 = 1.14
0.64 + -1.6y + y2 = 1.14

Factor a perfect square on the left side:
(y + -0.8)(y + -0.8) = 1.14

Calculate the square root of the right side: 1.067707825

Break this problem into two subproblems by setting 
(y + -0.8) equal to 1.067707825 and -1.067707825.

Subproblem 1

y + -0.8 = 1.067707825 Simplifying y + -0.8 = 1.067707825 Reorder the terms: -0.8 + y = 1.067707825 Solving -0.8 + y = 1.067707825 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.8' to each side of the equation. -0.8 + 0.8 + y = 1.067707825 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + y = 1.067707825 + 0.8 y = 1.067707825 + 0.8 Combine like terms: 1.067707825 + 0.8 = 1.867707825 y = 1.867707825 Simplifying y = 1.867707825

Subproblem 2

y + -0.8 = -1.067707825 Simplifying y + -0.8 = -1.067707825 Reorder the terms: -0.8 + y = -1.067707825 Solving -0.8 + y = -1.067707825 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.8' to each side of the equation. -0.8 + 0.8 + y = -1.067707825 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + y = -1.067707825 + 0.8 y = -1.067707825 + 0.8 Combine like terms: -1.067707825 + 0.8 = -0.267707825 y = -0.267707825 Simplifying y = -0.267707825

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.867707825, -0.267707825}

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