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Simplifying -28y3 + 56y2 + 12y2 + 24y = 0 Reorder the terms: 24y + 56y2 + 12y2 + -28y3 = 0 Combine like terms: 56y2 + 12y2 = 68y2 24y + 68y2 + -28y3 = 0 Solving 24y + 68y2 + -28y3 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '4y'. 4y(6 + 17y + -7y2) = 0 Ignore the factor 4.Subproblem 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 '(6 + 17y + -7y2)' equal to zero and attempt to solve: Simplifying 6 + 17y + -7y2 = 0 Solving 6 + 17y + -7y2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -0.8571428571 + -2.428571429y + y2 = 0 Move the constant term to the right: Add '0.8571428571' to each side of the equation. -0.8571428571 + -2.428571429y + 0.8571428571 + y2 = 0 + 0.8571428571 Reorder the terms: -0.8571428571 + 0.8571428571 + -2.428571429y + y2 = 0 + 0.8571428571 Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000 0.0000000000 + -2.428571429y + y2 = 0 + 0.8571428571 -2.428571429y + y2 = 0 + 0.8571428571 Combine like terms: 0 + 0.8571428571 = 0.8571428571 -2.428571429y + y2 = 0.8571428571 The y term is -2.428571429y. Take half its coefficient (-1.214285715). Square it (1.474489798) and add it to both sides. Add '1.474489798' to each side of the equation. -2.428571429y + 1.474489798 + y2 = 0.8571428571 + 1.474489798 Reorder the terms: 1.474489798 + -2.428571429y + y2 = 0.8571428571 + 1.474489798 Combine like terms: 0.8571428571 + 1.474489798 = 2.3316326551 1.474489798 + -2.428571429y + y2 = 2.3316326551 Factor a perfect square on the left side: (y + -1.214285715)(y + -1.214285715) = 2.3316326551 Calculate the square root of the right side: 1.526968453 Break this problem into two subproblems by setting (y + -1.214285715) equal to 1.526968453 and -1.526968453.Subproblem 1
y + -1.214285715 = 1.526968453 Simplifying y + -1.214285715 = 1.526968453 Reorder the terms: -1.214285715 + y = 1.526968453 Solving -1.214285715 + y = 1.526968453 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.214285715' to each side of the equation. -1.214285715 + 1.214285715 + y = 1.526968453 + 1.214285715 Combine like terms: -1.214285715 + 1.214285715 = 0.000000000 0.000000000 + y = 1.526968453 + 1.214285715 y = 1.526968453 + 1.214285715 Combine like terms: 1.526968453 + 1.214285715 = 2.741254168 y = 2.741254168 Simplifying y = 2.741254168Subproblem 2
y + -1.214285715 = -1.526968453 Simplifying y + -1.214285715 = -1.526968453 Reorder the terms: -1.214285715 + y = -1.526968453 Solving -1.214285715 + y = -1.526968453 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.214285715' to each side of the equation. -1.214285715 + 1.214285715 + y = -1.526968453 + 1.214285715 Combine like terms: -1.214285715 + 1.214285715 = 0.000000000 0.000000000 + y = -1.526968453 + 1.214285715 y = -1.526968453 + 1.214285715 Combine like terms: -1.526968453 + 1.214285715 = -0.312682738 y = -0.312682738 Simplifying y = -0.312682738Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.741254168, -0.312682738}Solution
y = {0, 2.741254168, -0.312682738}
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