(3y+5)(2y-7)=180

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Solution for (3y+5)(2y-7)=180 equation:


Simplifying
(3y + 5)(2y + -7) = 180

Reorder the terms:
(5 + 3y)(2y + -7) = 180

Reorder the terms:
(5 + 3y)(-7 + 2y) = 180

Multiply (5 + 3y) * (-7 + 2y)
(5(-7 + 2y) + 3y * (-7 + 2y)) = 180
((-7 * 5 + 2y * 5) + 3y * (-7 + 2y)) = 180
((-35 + 10y) + 3y * (-7 + 2y)) = 180
(-35 + 10y + (-7 * 3y + 2y * 3y)) = 180
(-35 + 10y + (-21y + 6y2)) = 180

Combine like terms: 10y + -21y = -11y
(-35 + -11y + 6y2) = 180

Solving
-35 + -11y + 6y2 = 180

Solving for variable 'y'.

Reorder the terms:
-35 + -180 + -11y + 6y2 = 180 + -180

Combine like terms: -35 + -180 = -215
-215 + -11y + 6y2 = 180 + -180

Combine like terms: 180 + -180 = 0
-215 + -11y + 6y2 = 0

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

Divide each side by '6'.
-35.83333333 + -1.833333333y + y2 = 0

Move the constant term to the right:

Add '35.83333333' to each side of the equation.
-35.83333333 + -1.833333333y + 35.83333333 + y2 = 0 + 35.83333333

Reorder the terms:
-35.83333333 + 35.83333333 + -1.833333333y + y2 = 0 + 35.83333333

Combine like terms: -35.83333333 + 35.83333333 = 0.00000000
0.00000000 + -1.833333333y + y2 = 0 + 35.83333333
-1.833333333y + y2 = 0 + 35.83333333

Combine like terms: 0 + 35.83333333 = 35.83333333
-1.833333333y + y2 = 35.83333333

The y term is -1.833333333y.  Take half its coefficient (-0.9166666665).
Square it (0.8402777775) and add it to both sides.

Add '0.8402777775' to each side of the equation.
-1.833333333y + 0.8402777775 + y2 = 35.83333333 + 0.8402777775

Reorder the terms:
0.8402777775 + -1.833333333y + y2 = 35.83333333 + 0.8402777775

Combine like terms: 35.83333333 + 0.8402777775 = 36.6736111075
0.8402777775 + -1.833333333y + y2 = 36.6736111075

Factor a perfect square on the left side:
(y + -0.9166666665)(y + -0.9166666665) = 36.6736111075

Calculate the square root of the right side: 6.055874099

Break this problem into two subproblems by setting 
(y + -0.9166666665) equal to 6.055874099 and -6.055874099.

Subproblem 1

y + -0.9166666665 = 6.055874099 Simplifying y + -0.9166666665 = 6.055874099 Reorder the terms: -0.9166666665 + y = 6.055874099 Solving -0.9166666665 + y = 6.055874099 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.9166666665' to each side of the equation. -0.9166666665 + 0.9166666665 + y = 6.055874099 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + y = 6.055874099 + 0.9166666665 y = 6.055874099 + 0.9166666665 Combine like terms: 6.055874099 + 0.9166666665 = 6.9725407655 y = 6.9725407655 Simplifying y = 6.9725407655

Subproblem 2

y + -0.9166666665 = -6.055874099 Simplifying y + -0.9166666665 = -6.055874099 Reorder the terms: -0.9166666665 + y = -6.055874099 Solving -0.9166666665 + y = -6.055874099 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.9166666665' to each side of the equation. -0.9166666665 + 0.9166666665 + y = -6.055874099 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + y = -6.055874099 + 0.9166666665 y = -6.055874099 + 0.9166666665 Combine like terms: -6.055874099 + 0.9166666665 = -5.1392074325 y = -5.1392074325 Simplifying y = -5.1392074325

Solution

The solution to the problem is based on the solutions from the subproblems. y = {6.9725407655, -5.1392074325}

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