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Simplifying y3 + 2y2 + -5y + -6y = 0 Reorder the terms: -5y + -6y + 2y2 + y3 = 0 Combine like terms: -5y + -6y = -11y -11y + 2y2 + y3 = 0 Solving -11y + 2y2 + y3 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), 'y'. y(-11 + 2y + y2) = 0Subproblem 1
Set the factor 'y' equal to zero and attempt to solve: Simplifying y = 0 Solving y = 0 Move all terms containing y to the left, all other terms to the right. Simplifying y = 0Subproblem 2
Set the factor '(-11 + 2y + y2)' equal to zero and attempt to solve: Simplifying -11 + 2y + y2 = 0 Solving -11 + 2y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '11' to each side of the equation. -11 + 2y + 11 + y2 = 0 + 11 Reorder the terms: -11 + 11 + 2y + y2 = 0 + 11 Combine like terms: -11 + 11 = 0 0 + 2y + y2 = 0 + 11 2y + y2 = 0 + 11 Combine like terms: 0 + 11 = 11 2y + y2 = 11 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 = 11 + 1 Reorder the terms: 1 + 2y + y2 = 11 + 1 Combine like terms: 11 + 1 = 12 1 + 2y + y2 = 12 Factor a perfect square on the left side: (y + 1)(y + 1) = 12 Calculate the square root of the right side: 3.464101615 Break this problem into two subproblems by setting (y + 1) equal to 3.464101615 and -3.464101615.Subproblem 1
y + 1 = 3.464101615 Simplifying y + 1 = 3.464101615 Reorder the terms: 1 + y = 3.464101615 Solving 1 + y = 3.464101615 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 = 3.464101615 + -1 Combine like terms: 1 + -1 = 0 0 + y = 3.464101615 + -1 y = 3.464101615 + -1 Combine like terms: 3.464101615 + -1 = 2.464101615 y = 2.464101615 Simplifying y = 2.464101615Subproblem 2
y + 1 = -3.464101615 Simplifying y + 1 = -3.464101615 Reorder the terms: 1 + y = -3.464101615 Solving 1 + y = -3.464101615 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 = -3.464101615 + -1 Combine like terms: 1 + -1 = 0 0 + y = -3.464101615 + -1 y = -3.464101615 + -1 Combine like terms: -3.464101615 + -1 = -4.464101615 y = -4.464101615 Simplifying y = -4.464101615Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.464101615, -4.464101615}Solution
y = {0, 2.464101615, -4.464101615}
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