5y^2=3y+4

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Solution for 5y^2=3y+4 equation:


Simplifying
5y2 = 3y + 4

Reorder the terms:
5y2 = 4 + 3y

Solving
5y2 = 4 + 3y

Solving for variable 'y'.

Reorder the terms:
-4 + -3y + 5y2 = 4 + 3y + -4 + -3y

Reorder the terms:
-4 + -3y + 5y2 = 4 + -4 + 3y + -3y

Combine like terms: 4 + -4 = 0
-4 + -3y + 5y2 = 0 + 3y + -3y
-4 + -3y + 5y2 = 3y + -3y

Combine like terms: 3y + -3y = 0
-4 + -3y + 5y2 = 0

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

Divide each side by '5'.
-0.8 + -0.6y + y2 = 0

Move the constant term to the right:

Add '0.8' to each side of the equation.
-0.8 + -0.6y + 0.8 + y2 = 0 + 0.8

Reorder the terms:
-0.8 + 0.8 + -0.6y + y2 = 0 + 0.8

Combine like terms: -0.8 + 0.8 = 0.0
0.0 + -0.6y + y2 = 0 + 0.8
-0.6y + y2 = 0 + 0.8

Combine like terms: 0 + 0.8 = 0.8
-0.6y + y2 = 0.8

The y term is -0.6y.  Take half its coefficient (-0.3).
Square it (0.09) and add it to both sides.

Add '0.09' to each side of the equation.
-0.6y + 0.09 + y2 = 0.8 + 0.09

Reorder the terms:
0.09 + -0.6y + y2 = 0.8 + 0.09

Combine like terms: 0.8 + 0.09 = 0.89
0.09 + -0.6y + y2 = 0.89

Factor a perfect square on the left side:
(y + -0.3)(y + -0.3) = 0.89

Calculate the square root of the right side: 0.943398113

Break this problem into two subproblems by setting 
(y + -0.3) equal to 0.943398113 and -0.943398113.

Subproblem 1

y + -0.3 = 0.943398113 Simplifying y + -0.3 = 0.943398113 Reorder the terms: -0.3 + y = 0.943398113 Solving -0.3 + y = 0.943398113 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + y = 0.943398113 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + y = 0.943398113 + 0.3 y = 0.943398113 + 0.3 Combine like terms: 0.943398113 + 0.3 = 1.243398113 y = 1.243398113 Simplifying y = 1.243398113

Subproblem 2

y + -0.3 = -0.943398113 Simplifying y + -0.3 = -0.943398113 Reorder the terms: -0.3 + y = -0.943398113 Solving -0.3 + y = -0.943398113 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + y = -0.943398113 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + y = -0.943398113 + 0.3 y = -0.943398113 + 0.3 Combine like terms: -0.943398113 + 0.3 = -0.643398113 y = -0.643398113 Simplifying y = -0.643398113

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.243398113, -0.643398113}

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