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Simplifying y(15 + 5y) = 415 (15 * y + 5y * y) = 415 (15y + 5y2) = 415 Solving 15y + 5y2 = 415 Solving for variable 'y'. Reorder the terms: -415 + 15y + 5y2 = 415 + -415 Combine like terms: 415 + -415 = 0 -415 + 15y + 5y2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-83 + 3y + y2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-83 + 3y + y2)' equal to zero and attempt to solve: Simplifying -83 + 3y + y2 = 0 Solving -83 + 3y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '83' to each side of the equation. -83 + 3y + 83 + y2 = 0 + 83 Reorder the terms: -83 + 83 + 3y + y2 = 0 + 83 Combine like terms: -83 + 83 = 0 0 + 3y + y2 = 0 + 83 3y + y2 = 0 + 83 Combine like terms: 0 + 83 = 83 3y + y2 = 83 The y term is 3y. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3y + 2.25 + y2 = 83 + 2.25 Reorder the terms: 2.25 + 3y + y2 = 83 + 2.25 Combine like terms: 83 + 2.25 = 85.25 2.25 + 3y + y2 = 85.25 Factor a perfect square on the left side: (y + 1.5)(y + 1.5) = 85.25 Calculate the square root of the right side: 9.233092656 Break this problem into two subproblems by setting (y + 1.5) equal to 9.233092656 and -9.233092656.Subproblem 1
y + 1.5 = 9.233092656 Simplifying y + 1.5 = 9.233092656 Reorder the terms: 1.5 + y = 9.233092656 Solving 1.5 + y = 9.233092656 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = 9.233092656 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = 9.233092656 + -1.5 y = 9.233092656 + -1.5 Combine like terms: 9.233092656 + -1.5 = 7.733092656 y = 7.733092656 Simplifying y = 7.733092656Subproblem 2
y + 1.5 = -9.233092656 Simplifying y + 1.5 = -9.233092656 Reorder the terms: 1.5 + y = -9.233092656 Solving 1.5 + y = -9.233092656 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = -9.233092656 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = -9.233092656 + -1.5 y = -9.233092656 + -1.5 Combine like terms: -9.233092656 + -1.5 = -10.733092656 y = -10.733092656 Simplifying y = -10.733092656Solution
The solution to the problem is based on the solutions from the subproblems. y = {7.733092656, -10.733092656}Solution
y = {7.733092656, -10.733092656}
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