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Simplifying y2 + 13y = 40 Reorder the terms: 13y + y2 = 40 Solving 13y + y2 = 40 Solving for variable 'y'. Reorder the terms: -40 + 13y + y2 = 40 + -40 Combine like terms: 40 + -40 = 0 -40 + 13y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '40' to each side of the equation. -40 + 13y + 40 + y2 = 0 + 40 Reorder the terms: -40 + 40 + 13y + y2 = 0 + 40 Combine like terms: -40 + 40 = 0 0 + 13y + y2 = 0 + 40 13y + y2 = 0 + 40 Combine like terms: 0 + 40 = 40 13y + y2 = 40 The y term is 13y. Take half its coefficient (6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. 13y + 42.25 + y2 = 40 + 42.25 Reorder the terms: 42.25 + 13y + y2 = 40 + 42.25 Combine like terms: 40 + 42.25 = 82.25 42.25 + 13y + y2 = 82.25 Factor a perfect square on the left side: (y + 6.5)(y + 6.5) = 82.25 Calculate the square root of the right side: 9.069178574 Break this problem into two subproblems by setting (y + 6.5) equal to 9.069178574 and -9.069178574.Subproblem 1
y + 6.5 = 9.069178574 Simplifying y + 6.5 = 9.069178574 Reorder the terms: 6.5 + y = 9.069178574 Solving 6.5 + y = 9.069178574 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + y = 9.069178574 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + y = 9.069178574 + -6.5 y = 9.069178574 + -6.5 Combine like terms: 9.069178574 + -6.5 = 2.569178574 y = 2.569178574 Simplifying y = 2.569178574Subproblem 2
y + 6.5 = -9.069178574 Simplifying y + 6.5 = -9.069178574 Reorder the terms: 6.5 + y = -9.069178574 Solving 6.5 + y = -9.069178574 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + y = -9.069178574 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + y = -9.069178574 + -6.5 y = -9.069178574 + -6.5 Combine like terms: -9.069178574 + -6.5 = -15.569178574 y = -15.569178574 Simplifying y = -15.569178574Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.569178574, -15.569178574}
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