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Simplifying (5 + y) * y = 3 Reorder the terms for easier multiplication: y(5 + y) = 3 (5 * y + y * y) = 3 (5y + y2) = 3 Solving 5y + y2 = 3 Solving for variable 'y'. Reorder the terms: -3 + 5y + y2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + 5y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + 5y + 3 + y2 = 0 + 3 Reorder the terms: -3 + 3 + 5y + y2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 5y + y2 = 0 + 3 5y + y2 = 0 + 3 Combine like terms: 0 + 3 = 3 5y + y2 = 3 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 = 3 + 6.25 Reorder the terms: 6.25 + 5y + y2 = 3 + 6.25 Combine like terms: 3 + 6.25 = 9.25 6.25 + 5y + y2 = 9.25 Factor a perfect square on the left side: (y + 2.5)(y + 2.5) = 9.25 Calculate the square root of the right side: 3.041381265 Break this problem into two subproblems by setting (y + 2.5) equal to 3.041381265 and -3.041381265.Subproblem 1
y + 2.5 = 3.041381265 Simplifying y + 2.5 = 3.041381265 Reorder the terms: 2.5 + y = 3.041381265 Solving 2.5 + y = 3.041381265 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 = 3.041381265 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + y = 3.041381265 + -2.5 y = 3.041381265 + -2.5 Combine like terms: 3.041381265 + -2.5 = 0.541381265 y = 0.541381265 Simplifying y = 0.541381265Subproblem 2
y + 2.5 = -3.041381265 Simplifying y + 2.5 = -3.041381265 Reorder the terms: 2.5 + y = -3.041381265 Solving 2.5 + y = -3.041381265 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 = -3.041381265 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + y = -3.041381265 + -2.5 y = -3.041381265 + -2.5 Combine like terms: -3.041381265 + -2.5 = -5.541381265 y = -5.541381265 Simplifying y = -5.541381265Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.541381265, -5.541381265}
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