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Simplifying y2 + 3y + 10 = 90 Reorder the terms: 10 + 3y + y2 = 90 Solving 10 + 3y + y2 = 90 Solving for variable 'y'. Reorder the terms: 10 + -90 + 3y + y2 = 90 + -90 Combine like terms: 10 + -90 = -80 -80 + 3y + y2 = 90 + -90 Combine like terms: 90 + -90 = 0 -80 + 3y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '80' to each side of the equation. -80 + 3y + 80 + y2 = 0 + 80 Reorder the terms: -80 + 80 + 3y + y2 = 0 + 80 Combine like terms: -80 + 80 = 0 0 + 3y + y2 = 0 + 80 3y + y2 = 0 + 80 Combine like terms: 0 + 80 = 80 3y + y2 = 80 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 = 80 + 2.25 Reorder the terms: 2.25 + 3y + y2 = 80 + 2.25 Combine like terms: 80 + 2.25 = 82.25 2.25 + 3y + y2 = 82.25 Factor a perfect square on the left side: (y + 1.5)(y + 1.5) = 82.25 Calculate the square root of the right side: 9.069178574 Break this problem into two subproblems by setting (y + 1.5) equal to 9.069178574 and -9.069178574.Subproblem 1
y + 1.5 = 9.069178574 Simplifying y + 1.5 = 9.069178574 Reorder the terms: 1.5 + y = 9.069178574 Solving 1.5 + y = 9.069178574 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 = 9.069178574 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = 9.069178574 + -1.5 y = 9.069178574 + -1.5 Combine like terms: 9.069178574 + -1.5 = 7.569178574 y = 7.569178574 Simplifying y = 7.569178574Subproblem 2
y + 1.5 = -9.069178574 Simplifying y + 1.5 = -9.069178574 Reorder the terms: 1.5 + y = -9.069178574 Solving 1.5 + y = -9.069178574 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 = -9.069178574 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = -9.069178574 + -1.5 y = -9.069178574 + -1.5 Combine like terms: -9.069178574 + -1.5 = -10.569178574 y = -10.569178574 Simplifying y = -10.569178574Solution
The solution to the problem is based on the solutions from the subproblems. y = {7.569178574, -10.569178574}
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