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 = 0.14
0.64 + -1.6y + y2 = 0.14

Factor a perfect square on the left side:
(y + -0.8)(y + -0.8) = 0.14

Calculate the square root of the right side: 0.374165739

Break this problem into two subproblems by setting 
(y + -0.8) equal to 0.374165739 and -0.374165739.

Subproblem 1

y + -0.8 = 0.374165739 Simplifying y + -0.8 = 0.374165739 Reorder the terms: -0.8 + y = 0.374165739 Solving -0.8 + y = 0.374165739 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 = 0.374165739 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + y = 0.374165739 + 0.8 y = 0.374165739 + 0.8 Combine like terms: 0.374165739 + 0.8 = 1.174165739 y = 1.174165739 Simplifying y = 1.174165739

Subproblem 2

y + -0.8 = -0.374165739 Simplifying y + -0.8 = -0.374165739 Reorder the terms: -0.8 + y = -0.374165739 Solving -0.8 + y = -0.374165739 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 = -0.374165739 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + y = -0.374165739 + 0.8 y = -0.374165739 + 0.8 Combine like terms: -0.374165739 + 0.8 = 0.425834261 y = 0.425834261 Simplifying y = 0.425834261

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.174165739, 0.425834261}

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