3y^2+5y=4

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


Simplifying
3y2 + 5y = 4

Reorder the terms:
5y + 3y2 = 4

Solving
5y + 3y2 = 4

Solving for variable 'y'.

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

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

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

Divide each side by '3'.
-1.333333333 + 1.666666667y + y2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + 1.666666667y + 1.333333333 + y2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + 1.666666667y + y2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + 1.666666667y + y2 = 0 + 1.333333333
1.666666667y + y2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
1.666666667y + y2 = 1.333333333

The y term is 1.666666667y.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667y + 0.6944444447 + y2 = 1.333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667y + y2 = 1.333333333 + 0.6944444447

Combine like terms: 1.333333333 + 0.6944444447 = 2.0277777777
0.6944444447 + 1.666666667y + y2 = 2.0277777777

Factor a perfect square on the left side:
(y + 0.8333333335)(y + 0.8333333335) = 2.0277777777

Calculate the square root of the right side: 1.424000624

Break this problem into two subproblems by setting 
(y + 0.8333333335) equal to 1.424000624 and -1.424000624.


Subproblem 1
y + 0.8333333335 = 1.424000624

Simplifying
y + 0.8333333335 = 1.424000624

Reorder the terms:
0.8333333335 + y = 1.424000624

Solving
0.8333333335 + y = 1.424000624

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + y = 1.424000624 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + y = 1.424000624 + -0.8333333335
y = 1.424000624 + -0.8333333335

Combine like terms: 1.424000624 + -0.8333333335 = 0.5906672905
y = 0.5906672905

Simplifying
y = 0.5906672905


Subproblem 2
y + 0.8333333335 = -1.424000624

Simplifying
y + 0.8333333335 = -1.424000624

Reorder the terms:
0.8333333335 + y = -1.424000624

Solving
0.8333333335 + y = -1.424000624

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + y = -1.424000624 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + y = -1.424000624 + -0.8333333335
y = -1.424000624 + -0.8333333335

Combine like terms: -1.424000624 + -0.8333333335 = -2.2573339575
y = -2.2573339575

Simplifying
y = -2.2573339575


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.5906672905, -2.2573339575}

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