2y^2+3y+5=8

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Solution for 2y^2+3y+5=8 equation:


Simplifying
2y2 + 3y + 5 = 8

Reorder the terms:
5 + 3y + 2y2 = 8

Solving
5 + 3y + 2y2 = 8

Solving for variable 'y'.

Reorder the terms:
5 + -8 + 3y + 2y2 = 8 + -8

Combine like terms: 5 + -8 = -3
-3 + 3y + 2y2 = 8 + -8

Combine like terms: 8 + -8 = 0
-3 + 3y + 2y2 = 0

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

Divide each side by '2'.
-1.5 + 1.5y + y2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + 1.5y + 1.5 + y2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + 1.5y + y2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + 1.5y + y2 = 0 + 1.5
1.5y + y2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
1.5y + y2 = 1.5

The y term is 1.5y.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5y + 0.5625 + y2 = 1.5 + 0.5625

Reorder the terms:
0.5625 + 1.5y + y2 = 1.5 + 0.5625

Combine like terms: 1.5 + 0.5625 = 2.0625
0.5625 + 1.5y + y2 = 2.0625

Factor a perfect square on the left side:
(y + 0.75)(y + 0.75) = 2.0625

Calculate the square root of the right side: 1.436140662

Break this problem into two subproblems by setting 
(y + 0.75) equal to 1.436140662 and -1.436140662.

Subproblem 1

y + 0.75 = 1.436140662 Simplifying y + 0.75 = 1.436140662 Reorder the terms: 0.75 + y = 1.436140662 Solving 0.75 + y = 1.436140662 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = 1.436140662 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = 1.436140662 + -0.75 y = 1.436140662 + -0.75 Combine like terms: 1.436140662 + -0.75 = 0.686140662 y = 0.686140662 Simplifying y = 0.686140662

Subproblem 2

y + 0.75 = -1.436140662 Simplifying y + 0.75 = -1.436140662 Reorder the terms: 0.75 + y = -1.436140662 Solving 0.75 + y = -1.436140662 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = -1.436140662 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = -1.436140662 + -0.75 y = -1.436140662 + -0.75 Combine like terms: -1.436140662 + -0.75 = -2.186140662 y = -2.186140662 Simplifying y = -2.186140662

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.686140662, -2.186140662}

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