4y^2+4y=65

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Solution for 4y^2+4y=65 equation:


Simplifying
4y2 + 4y = 65

Reorder the terms:
4y + 4y2 = 65

Solving
4y + 4y2 = 65

Solving for variable 'y'.

Reorder the terms:
-65 + 4y + 4y2 = 65 + -65

Combine like terms: 65 + -65 = 0
-65 + 4y + 4y2 = 0

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

Divide each side by '4'.
-16.25 + y + y2 = 0

Move the constant term to the right:

Add '16.25' to each side of the equation.
-16.25 + y + 16.25 + y2 = 0 + 16.25

Reorder the terms:
-16.25 + 16.25 + y + y2 = 0 + 16.25

Combine like terms: -16.25 + 16.25 = 0.00
0.00 + y + y2 = 0 + 16.25
y + y2 = 0 + 16.25

Combine like terms: 0 + 16.25 = 16.25
y + y2 = 16.25

The y term is y.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + y2 = 16.25 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + y2 = 16.25 + 0.25

Combine like terms: 16.25 + 0.25 = 16.5
1.25 + y2 = 16.5

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 16.5

Calculate the square root of the right side: 4.062019202

Break this problem into two subproblems by setting 
(y + 0.5) equal to 4.062019202 and -4.062019202.

Subproblem 1

y + 0.5 = 4.062019202 Simplifying y + 0.5 = 4.062019202 Reorder the terms: 0.5 + y = 4.062019202 Solving 0.5 + y = 4.062019202 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = 4.062019202 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 4.062019202 + -0.5 y = 4.062019202 + -0.5 Combine like terms: 4.062019202 + -0.5 = 3.562019202 y = 3.562019202 Simplifying y = 3.562019202

Subproblem 2

y + 0.5 = -4.062019202 Simplifying y + 0.5 = -4.062019202 Reorder the terms: 0.5 + y = -4.062019202 Solving 0.5 + y = -4.062019202 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = -4.062019202 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -4.062019202 + -0.5 y = -4.062019202 + -0.5 Combine like terms: -4.062019202 + -0.5 = -4.562019202 y = -4.562019202 Simplifying y = -4.562019202

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.562019202, -4.562019202}

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