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