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Simplifying 5y + 23 = y2 Reorder the terms: 23 + 5y = y2 Solving 23 + 5y = y2 Solving for variable 'y'. Combine like terms: y2 + -1y2 = 0 23 + 5y + -1y2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -23 + -5y + y2 = 0 Move the constant term to the right: Add '23' to each side of the equation. -23 + -5y + 23 + y2 = 0 + 23 Reorder the terms: -23 + 23 + -5y + y2 = 0 + 23 Combine like terms: -23 + 23 = 0 0 + -5y + y2 = 0 + 23 -5y + y2 = 0 + 23 Combine like terms: 0 + 23 = 23 -5y + y2 = 23 The y term is -5y. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5y + 6.25 + y2 = 23 + 6.25 Reorder the terms: 6.25 + -5y + y2 = 23 + 6.25 Combine like terms: 23 + 6.25 = 29.25 6.25 + -5y + y2 = 29.25 Factor a perfect square on the left side: (y + -2.5)(y + -2.5) = 29.25 Calculate the square root of the right side: 5.408326913 Break this problem into two subproblems by setting (y + -2.5) equal to 5.408326913 and -5.408326913.Subproblem 1
y + -2.5 = 5.408326913 Simplifying y + -2.5 = 5.408326913 Reorder the terms: -2.5 + y = 5.408326913 Solving -2.5 + y = 5.408326913 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + y = 5.408326913 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + y = 5.408326913 + 2.5 y = 5.408326913 + 2.5 Combine like terms: 5.408326913 + 2.5 = 7.908326913 y = 7.908326913 Simplifying y = 7.908326913Subproblem 2
y + -2.5 = -5.408326913 Simplifying y + -2.5 = -5.408326913 Reorder the terms: -2.5 + y = -5.408326913 Solving -2.5 + y = -5.408326913 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + y = -5.408326913 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + y = -5.408326913 + 2.5 y = -5.408326913 + 2.5 Combine like terms: -5.408326913 + 2.5 = -2.908326913 y = -2.908326913 Simplifying y = -2.908326913Solution
The solution to the problem is based on the solutions from the subproblems. y = {7.908326913, -2.908326913}
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