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Simplifying 5 + 3y + -1y2 + 2 + -5y = 0 Reorder the terms: 5 + 2 + 3y + -5y + -1y2 = 0 Combine like terms: 5 + 2 = 7 7 + 3y + -5y + -1y2 = 0 Combine like terms: 3y + -5y = -2y 7 + -2y + -1y2 = 0 Solving 7 + -2y + -1y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -7 + 2y + y2 = 0 Move the constant term to the right: Add '7' to each side of the equation. -7 + 2y + 7 + y2 = 0 + 7 Reorder the terms: -7 + 7 + 2y + y2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + 2y + y2 = 0 + 7 2y + y2 = 0 + 7 Combine like terms: 0 + 7 = 7 2y + y2 = 7 The y term is 2y. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2y + 1 + y2 = 7 + 1 Reorder the terms: 1 + 2y + y2 = 7 + 1 Combine like terms: 7 + 1 = 8 1 + 2y + y2 = 8 Factor a perfect square on the left side: (y + 1)(y + 1) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (y + 1) equal to 2.828427125 and -2.828427125.Subproblem 1
y + 1 = 2.828427125 Simplifying y + 1 = 2.828427125 Reorder the terms: 1 + y = 2.828427125 Solving 1 + y = 2.828427125 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = 2.828427125 + -1 Combine like terms: 1 + -1 = 0 0 + y = 2.828427125 + -1 y = 2.828427125 + -1 Combine like terms: 2.828427125 + -1 = 1.828427125 y = 1.828427125 Simplifying y = 1.828427125Subproblem 2
y + 1 = -2.828427125 Simplifying y + 1 = -2.828427125 Reorder the terms: 1 + y = -2.828427125 Solving 1 + y = -2.828427125 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = -2.828427125 + -1 Combine like terms: 1 + -1 = 0 0 + y = -2.828427125 + -1 y = -2.828427125 + -1 Combine like terms: -2.828427125 + -1 = -3.828427125 y = -3.828427125 Simplifying y = -3.828427125Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.828427125, -3.828427125}
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