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Simplifying (y + 3) * y = 14 Reorder the terms: (3 + y) * y = 14 Reorder the terms for easier multiplication: y(3 + y) = 14 (3 * y + y * y) = 14 (3y + y2) = 14 Solving 3y + y2 = 14 Solving for variable 'y'. Reorder the terms: -14 + 3y + y2 = 14 + -14 Combine like terms: 14 + -14 = 0 -14 + 3y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '14' to each side of the equation. -14 + 3y + 14 + y2 = 0 + 14 Reorder the terms: -14 + 14 + 3y + y2 = 0 + 14 Combine like terms: -14 + 14 = 0 0 + 3y + y2 = 0 + 14 3y + y2 = 0 + 14 Combine like terms: 0 + 14 = 14 3y + y2 = 14 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 = 14 + 2.25 Reorder the terms: 2.25 + 3y + y2 = 14 + 2.25 Combine like terms: 14 + 2.25 = 16.25 2.25 + 3y + y2 = 16.25 Factor a perfect square on the left side: (y + 1.5)(y + 1.5) = 16.25 Calculate the square root of the right side: 4.031128874 Break this problem into two subproblems by setting (y + 1.5) equal to 4.031128874 and -4.031128874.Subproblem 1
y + 1.5 = 4.031128874 Simplifying y + 1.5 = 4.031128874 Reorder the terms: 1.5 + y = 4.031128874 Solving 1.5 + y = 4.031128874 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 = 4.031128874 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = 4.031128874 + -1.5 y = 4.031128874 + -1.5 Combine like terms: 4.031128874 + -1.5 = 2.531128874 y = 2.531128874 Simplifying y = 2.531128874Subproblem 2
y + 1.5 = -4.031128874 Simplifying y + 1.5 = -4.031128874 Reorder the terms: 1.5 + y = -4.031128874 Solving 1.5 + y = -4.031128874 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 = -4.031128874 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = -4.031128874 + -1.5 y = -4.031128874 + -1.5 Combine like terms: -4.031128874 + -1.5 = -5.531128874 y = -5.531128874 Simplifying y = -5.531128874Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.531128874, -5.531128874}
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