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Simplifying 2.5(y2 + 2y + -3) + 1.4y2 + 3.6 = 0 Reorder the terms: 2.5(-3 + 2y + y2) + 1.4y2 + 3.6 = 0 (-3 * 2.5 + 2y * 2.5 + y2 * 2.5) + 1.4y2 + 3.6 = 0 (-7.5 + 5y + 2.5y2) + 1.4y2 + 3.6 = 0 Reorder the terms: -7.5 + 3.6 + 5y + 2.5y2 + 1.4y2 = 0 Combine like terms: -7.5 + 3.6 = -3.9 -3.9 + 5y + 2.5y2 + 1.4y2 = 0 Combine like terms: 2.5y2 + 1.4y2 = 3.9y2 -3.9 + 5y + 3.9y2 = 0 Solving -3.9 + 5y + 3.9y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 3.9 the coefficient of the squared term: Divide each side by '3.9'. -1 + 1.282051282y + y2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 1.282051282y + 1 + y2 = 0 + 1 Reorder the terms: -1 + 1 + 1.282051282y + y2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 1.282051282y + y2 = 0 + 1 1.282051282y + y2 = 0 + 1 Combine like terms: 0 + 1 = 1 1.282051282y + y2 = 1 The y term is 1.282051282y. Take half its coefficient (0.641025641). Square it (0.4109138724) and add it to both sides. Add '0.4109138724' to each side of the equation. 1.282051282y + 0.4109138724 + y2 = 1 + 0.4109138724 Reorder the terms: 0.4109138724 + 1.282051282y + y2 = 1 + 0.4109138724 Combine like terms: 1 + 0.4109138724 = 1.4109138724 0.4109138724 + 1.282051282y + y2 = 1.4109138724 Factor a perfect square on the left side: (y + 0.641025641)(y + 0.641025641) = 1.4109138724 Calculate the square root of the right side: 1.187818956 Break this problem into two subproblems by setting (y + 0.641025641) equal to 1.187818956 and -1.187818956.Subproblem 1
y + 0.641025641 = 1.187818956 Simplifying y + 0.641025641 = 1.187818956 Reorder the terms: 0.641025641 + y = 1.187818956 Solving 0.641025641 + y = 1.187818956 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.641025641' to each side of the equation. 0.641025641 + -0.641025641 + y = 1.187818956 + -0.641025641 Combine like terms: 0.641025641 + -0.641025641 = 0.000000000 0.000000000 + y = 1.187818956 + -0.641025641 y = 1.187818956 + -0.641025641 Combine like terms: 1.187818956 + -0.641025641 = 0.546793315 y = 0.546793315 Simplifying y = 0.546793315Subproblem 2
y + 0.641025641 = -1.187818956 Simplifying y + 0.641025641 = -1.187818956 Reorder the terms: 0.641025641 + y = -1.187818956 Solving 0.641025641 + y = -1.187818956 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.641025641' to each side of the equation. 0.641025641 + -0.641025641 + y = -1.187818956 + -0.641025641 Combine like terms: 0.641025641 + -0.641025641 = 0.000000000 0.000000000 + y = -1.187818956 + -0.641025641 y = -1.187818956 + -0.641025641 Combine like terms: -1.187818956 + -0.641025641 = -1.828844597 y = -1.828844597 Simplifying y = -1.828844597Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.546793315, -1.828844597}
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