5y+3y(y+2)=54

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Solution for 5y+3y(y+2)=54 equation:


Simplifying
5y + 3y(y + 2) = 54

Reorder the terms:
5y + 3y(2 + y) = 54
5y + (2 * 3y + y * 3y) = 54
5y + (6y + 3y2) = 54

Combine like terms: 5y + 6y = 11y
11y + 3y2 = 54

Solving
11y + 3y2 = 54

Solving for variable 'y'.

Reorder the terms:
-54 + 11y + 3y2 = 54 + -54

Combine like terms: 54 + -54 = 0
-54 + 11y + 3y2 = 0

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

Divide each side by '3'.
-18 + 3.666666667y + y2 = 0

Move the constant term to the right:

Add '18' to each side of the equation.
-18 + 3.666666667y + 18 + y2 = 0 + 18

Reorder the terms:
-18 + 18 + 3.666666667y + y2 = 0 + 18

Combine like terms: -18 + 18 = 0
0 + 3.666666667y + y2 = 0 + 18
3.666666667y + y2 = 0 + 18

Combine like terms: 0 + 18 = 18
3.666666667y + y2 = 18

The y term is 3.666666667y.  Take half its coefficient (1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
3.666666667y + 3.361111114 + y2 = 18 + 3.361111114

Reorder the terms:
3.361111114 + 3.666666667y + y2 = 18 + 3.361111114

Combine like terms: 18 + 3.361111114 = 21.361111114
3.361111114 + 3.666666667y + y2 = 21.361111114

Factor a perfect square on the left side:
(y + 1.833333334)(y + 1.833333334) = 21.361111114

Calculate the square root of the right side: 4.621808208

Break this problem into two subproblems by setting 
(y + 1.833333334) equal to 4.621808208 and -4.621808208.

Subproblem 1

y + 1.833333334 = 4.621808208 Simplifying y + 1.833333334 = 4.621808208 Reorder the terms: 1.833333334 + y = 4.621808208 Solving 1.833333334 + y = 4.621808208 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + y = 4.621808208 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + y = 4.621808208 + -1.833333334 y = 4.621808208 + -1.833333334 Combine like terms: 4.621808208 + -1.833333334 = 2.788474874 y = 2.788474874 Simplifying y = 2.788474874

Subproblem 2

y + 1.833333334 = -4.621808208 Simplifying y + 1.833333334 = -4.621808208 Reorder the terms: 1.833333334 + y = -4.621808208 Solving 1.833333334 + y = -4.621808208 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + y = -4.621808208 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + y = -4.621808208 + -1.833333334 y = -4.621808208 + -1.833333334 Combine like terms: -4.621808208 + -1.833333334 = -6.455141542 y = -6.455141542 Simplifying y = -6.455141542

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.788474874, -6.455141542}

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