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Simplifying y2 + 1.5y = 54 Reorder the terms: 1.5y + y2 = 54 Solving 1.5y + y2 = 54 Solving for variable 'y'. Reorder the terms: -54 + 1.5y + y2 = 54 + -54 Combine like terms: 54 + -54 = 0 -54 + 1.5y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '54' to each side of the equation. -54 + 1.5y + 54 + y2 = 0 + 54 Reorder the terms: -54 + 54 + 1.5y + y2 = 0 + 54 Combine like terms: -54 + 54 = 0 0 + 1.5y + y2 = 0 + 54 1.5y + y2 = 0 + 54 Combine like terms: 0 + 54 = 54 1.5y + y2 = 54 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 = 54 + 0.5625 Reorder the terms: 0.5625 + 1.5y + y2 = 54 + 0.5625 Combine like terms: 54 + 0.5625 = 54.5625 0.5625 + 1.5y + y2 = 54.5625 Factor a perfect square on the left side: (y + 0.75)(y + 0.75) = 54.5625 Calculate the square root of the right side: 7.386643351 Break this problem into two subproblems by setting (y + 0.75) equal to 7.386643351 and -7.386643351.Subproblem 1
y + 0.75 = 7.386643351 Simplifying y + 0.75 = 7.386643351 Reorder the terms: 0.75 + y = 7.386643351 Solving 0.75 + y = 7.386643351 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 = 7.386643351 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = 7.386643351 + -0.75 y = 7.386643351 + -0.75 Combine like terms: 7.386643351 + -0.75 = 6.636643351 y = 6.636643351 Simplifying y = 6.636643351Subproblem 2
y + 0.75 = -7.386643351 Simplifying y + 0.75 = -7.386643351 Reorder the terms: 0.75 + y = -7.386643351 Solving 0.75 + y = -7.386643351 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 = -7.386643351 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = -7.386643351 + -0.75 y = -7.386643351 + -0.75 Combine like terms: -7.386643351 + -0.75 = -8.136643351 y = -8.136643351 Simplifying y = -8.136643351Solution
The solution to the problem is based on the solutions from the subproblems. y = {6.636643351, -8.136643351}
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