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Simplifying y + 111 = y * y Reorder the terms: 111 + y = y * y Multiply y * y 111 + y = y2 Solving 111 + y = y2 Solving for variable 'y'. Combine like terms: y2 + -1y2 = 0 111 + y + -1y2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -111 + -1y + y2 = 0 Move the constant term to the right: Add '111' to each side of the equation. -111 + -1y + 111 + y2 = 0 + 111 Reorder the terms: -111 + 111 + -1y + y2 = 0 + 111 Combine like terms: -111 + 111 = 0 0 + -1y + y2 = 0 + 111 -1y + y2 = 0 + 111 Combine like terms: 0 + 111 = 111 -1y + y2 = 111 The y term is y. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1y + 0.25 + y2 = 111 + 0.25 Reorder the terms: 0.25 + -1y + y2 = 111 + 0.25 Combine like terms: 111 + 0.25 = 111.25 0.25 + -1y + y2 = 111.25 Factor a perfect square on the left side: (y + 0.5)(y + 0.5) = 111.25 Calculate the square root of the right side: 10.547511555 Break this problem into two subproblems by setting (y + 0.5) equal to 10.547511555 and -10.547511555.Subproblem 1
y + 0.5 = 10.547511555 Simplifying y + 0.5 = 10.547511555 Reorder the terms: 0.5 + y = 10.547511555 Solving 0.5 + y = 10.547511555 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = 10.547511555 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 10.547511555 + -0.5 y = 10.547511555 + -0.5 Combine like terms: 10.547511555 + -0.5 = 10.047511555 y = 10.047511555 Simplifying y = 10.047511555Subproblem 2
y + 0.5 = -10.547511555 Simplifying y + 0.5 = -10.547511555 Reorder the terms: 0.5 + y = -10.547511555 Solving 0.5 + y = -10.547511555 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = -10.547511555 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -10.547511555 + -0.5 y = -10.547511555 + -0.5 Combine like terms: -10.547511555 + -0.5 = -11.047511555 y = -11.047511555 Simplifying y = -11.047511555Solution
The solution to the problem is based on the solutions from the subproblems. y = {10.047511555, -11.047511555}
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