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Simplifying 4y2 + 11y + 35 + 2y2 + -2y + -85 = 180 Reorder the terms: 35 + -85 + 11y + -2y + 4y2 + 2y2 = 180 Combine like terms: 35 + -85 = -50 -50 + 11y + -2y + 4y2 + 2y2 = 180 Combine like terms: 11y + -2y = 9y -50 + 9y + 4y2 + 2y2 = 180 Combine like terms: 4y2 + 2y2 = 6y2 -50 + 9y + 6y2 = 180 Solving -50 + 9y + 6y2 = 180 Solving for variable 'y'. Reorder the terms: -50 + -180 + 9y + 6y2 = 180 + -180 Combine like terms: -50 + -180 = -230 -230 + 9y + 6y2 = 180 + -180 Combine like terms: 180 + -180 = 0 -230 + 9y + 6y2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -38.33333333 + 1.5y + y2 = 0 Move the constant term to the right: Add '38.33333333' to each side of the equation. -38.33333333 + 1.5y + 38.33333333 + y2 = 0 + 38.33333333 Reorder the terms: -38.33333333 + 38.33333333 + 1.5y + y2 = 0 + 38.33333333 Combine like terms: -38.33333333 + 38.33333333 = 0.00000000 0.00000000 + 1.5y + y2 = 0 + 38.33333333 1.5y + y2 = 0 + 38.33333333 Combine like terms: 0 + 38.33333333 = 38.33333333 1.5y + y2 = 38.33333333 The y term is 1.5y. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5y + 0.5625 + y2 = 38.33333333 + 0.5625 Reorder the terms: 0.5625 + 1.5y + y2 = 38.33333333 + 0.5625 Combine like terms: 38.33333333 + 0.5625 = 38.89583333 0.5625 + 1.5y + y2 = 38.89583333 Factor a perfect square on the left side: (y + 0.75)(y + 0.75) = 38.89583333 Calculate the square root of the right side: 6.236652414 Break this problem into two subproblems by setting (y + 0.75) equal to 6.236652414 and -6.236652414.Subproblem 1
y + 0.75 = 6.236652414 Simplifying y + 0.75 = 6.236652414 Reorder the terms: 0.75 + y = 6.236652414 Solving 0.75 + y = 6.236652414 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = 6.236652414 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = 6.236652414 + -0.75 y = 6.236652414 + -0.75 Combine like terms: 6.236652414 + -0.75 = 5.486652414 y = 5.486652414 Simplifying y = 5.486652414Subproblem 2
y + 0.75 = -6.236652414 Simplifying y + 0.75 = -6.236652414 Reorder the terms: 0.75 + y = -6.236652414 Solving 0.75 + y = -6.236652414 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = -6.236652414 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = -6.236652414 + -0.75 y = -6.236652414 + -0.75 Combine like terms: -6.236652414 + -0.75 = -6.986652414 y = -6.986652414 Simplifying y = -6.986652414Solution
The solution to the problem is based on the solutions from the subproblems. y = {5.486652414, -6.986652414}
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