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Simplifying (5y + 1)(3y + -7) = 180 Reorder the terms: (1 + 5y)(3y + -7) = 180 Reorder the terms: (1 + 5y)(-7 + 3y) = 180 Multiply (1 + 5y) * (-7 + 3y) (1(-7 + 3y) + 5y * (-7 + 3y)) = 180 ((-7 * 1 + 3y * 1) + 5y * (-7 + 3y)) = 180 ((-7 + 3y) + 5y * (-7 + 3y)) = 180 (-7 + 3y + (-7 * 5y + 3y * 5y)) = 180 (-7 + 3y + (-35y + 15y2)) = 180 Combine like terms: 3y + -35y = -32y (-7 + -32y + 15y2) = 180 Solving -7 + -32y + 15y2 = 180 Solving for variable 'y'. Reorder the terms: -7 + -180 + -32y + 15y2 = 180 + -180 Combine like terms: -7 + -180 = -187 -187 + -32y + 15y2 = 180 + -180 Combine like terms: 180 + -180 = 0 -187 + -32y + 15y2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -12.46666667 + -2.133333333y + y2 = 0 Move the constant term to the right: Add '12.46666667' to each side of the equation. -12.46666667 + -2.133333333y + 12.46666667 + y2 = 0 + 12.46666667 Reorder the terms: -12.46666667 + 12.46666667 + -2.133333333y + y2 = 0 + 12.46666667 Combine like terms: -12.46666667 + 12.46666667 = 0.00000000 0.00000000 + -2.133333333y + y2 = 0 + 12.46666667 -2.133333333y + y2 = 0 + 12.46666667 Combine like terms: 0 + 12.46666667 = 12.46666667 -2.133333333y + y2 = 12.46666667 The y term is -2.133333333y. Take half its coefficient (-1.066666667). Square it (1.137777778) and add it to both sides. Add '1.137777778' to each side of the equation. -2.133333333y + 1.137777778 + y2 = 12.46666667 + 1.137777778 Reorder the terms: 1.137777778 + -2.133333333y + y2 = 12.46666667 + 1.137777778 Combine like terms: 12.46666667 + 1.137777778 = 13.604444448 1.137777778 + -2.133333333y + y2 = 13.604444448 Factor a perfect square on the left side: (y + -1.066666667)(y + -1.066666667) = 13.604444448 Calculate the square root of the right side: 3.688420319 Break this problem into two subproblems by setting (y + -1.066666667) equal to 3.688420319 and -3.688420319.Subproblem 1
y + -1.066666667 = 3.688420319 Simplifying y + -1.066666667 = 3.688420319 Reorder the terms: -1.066666667 + y = 3.688420319 Solving -1.066666667 + y = 3.688420319 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.066666667' to each side of the equation. -1.066666667 + 1.066666667 + y = 3.688420319 + 1.066666667 Combine like terms: -1.066666667 + 1.066666667 = 0.000000000 0.000000000 + y = 3.688420319 + 1.066666667 y = 3.688420319 + 1.066666667 Combine like terms: 3.688420319 + 1.066666667 = 4.755086986 y = 4.755086986 Simplifying y = 4.755086986Subproblem 2
y + -1.066666667 = -3.688420319 Simplifying y + -1.066666667 = -3.688420319 Reorder the terms: -1.066666667 + y = -3.688420319 Solving -1.066666667 + y = -3.688420319 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.066666667' to each side of the equation. -1.066666667 + 1.066666667 + y = -3.688420319 + 1.066666667 Combine like terms: -1.066666667 + 1.066666667 = 0.000000000 0.000000000 + y = -3.688420319 + 1.066666667 y = -3.688420319 + 1.066666667 Combine like terms: -3.688420319 + 1.066666667 = -2.621753652 y = -2.621753652 Simplifying y = -2.621753652Solution
The solution to the problem is based on the solutions from the subproblems. y = {4.755086986, -2.621753652}
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