2y+3=3y(y+7)

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Solution for 2y+3=3y(y+7) equation:


Simplifying
2y + 3 = 3y(y + 7)

Reorder the terms:
3 + 2y = 3y(y + 7)

Reorder the terms:
3 + 2y = 3y(7 + y)
3 + 2y = (7 * 3y + y * 3y)
3 + 2y = (21y + 3y2)

Solving
3 + 2y = 21y + 3y2

Solving for variable 'y'.

Combine like terms: 2y + -21y = -19y
3 + -19y + -3y2 = 21y + 3y2 + -21y + -3y2

Reorder the terms:
3 + -19y + -3y2 = 21y + -21y + 3y2 + -3y2

Combine like terms: 21y + -21y = 0
3 + -19y + -3y2 = 0 + 3y2 + -3y2
3 + -19y + -3y2 = 3y2 + -3y2

Combine like terms: 3y2 + -3y2 = 0
3 + -19y + -3y2 = 0

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

Divide each side by '-3'.
-1 + 6.333333333y + y2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 6.333333333y + 1 + y2 = 0 + 1

Reorder the terms:
-1 + 1 + 6.333333333y + y2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 6.333333333y + y2 = 0 + 1
6.333333333y + y2 = 0 + 1

Combine like terms: 0 + 1 = 1
6.333333333y + y2 = 1

The y term is 6.333333333y.  Take half its coefficient (3.166666667).
Square it (10.02777778) and add it to both sides.

Add '10.02777778' to each side of the equation.
6.333333333y + 10.02777778 + y2 = 1 + 10.02777778

Reorder the terms:
10.02777778 + 6.333333333y + y2 = 1 + 10.02777778

Combine like terms: 1 + 10.02777778 = 11.02777778
10.02777778 + 6.333333333y + y2 = 11.02777778

Factor a perfect square on the left side:
(y + 3.166666667)(y + 3.166666667) = 11.02777778

Calculate the square root of the right side: 3.320809808

Break this problem into two subproblems by setting 
(y + 3.166666667) equal to 3.320809808 and -3.320809808.

Subproblem 1

y + 3.166666667 = 3.320809808 Simplifying y + 3.166666667 = 3.320809808 Reorder the terms: 3.166666667 + y = 3.320809808 Solving 3.166666667 + y = 3.320809808 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-3.166666667' to each side of the equation. 3.166666667 + -3.166666667 + y = 3.320809808 + -3.166666667 Combine like terms: 3.166666667 + -3.166666667 = 0.000000000 0.000000000 + y = 3.320809808 + -3.166666667 y = 3.320809808 + -3.166666667 Combine like terms: 3.320809808 + -3.166666667 = 0.154143141 y = 0.154143141 Simplifying y = 0.154143141

Subproblem 2

y + 3.166666667 = -3.320809808 Simplifying y + 3.166666667 = -3.320809808 Reorder the terms: 3.166666667 + y = -3.320809808 Solving 3.166666667 + y = -3.320809808 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-3.166666667' to each side of the equation. 3.166666667 + -3.166666667 + y = -3.320809808 + -3.166666667 Combine like terms: 3.166666667 + -3.166666667 = 0.000000000 0.000000000 + y = -3.320809808 + -3.166666667 y = -3.320809808 + -3.166666667 Combine like terms: -3.320809808 + -3.166666667 = -6.487476475 y = -6.487476475 Simplifying y = -6.487476475

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.154143141, -6.487476475}

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