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Simplifying 6y4 + -12y3 + 4y2 = 0 Reorder the terms: 4y2 + -12y3 + 6y4 = 0 Solving 4y2 + -12y3 + 6y4 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '2y2'. 2y2(2 + -6y + 3y2) = 0 Ignore the factor 2.Subproblem 1
Set the factor 'y2' equal to zero and attempt to solve: Simplifying y2 = 0 Solving y2 = 0 Move all terms containing y to the left, all other terms to the right. Simplifying y2 = 0 Take the square root of each side: y = {0}Subproblem 2
Set the factor '(2 + -6y + 3y2)' equal to zero and attempt to solve: Simplifying 2 + -6y + 3y2 = 0 Solving 2 + -6y + 3y2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.6666666667 + -2y + y2 = 0 Move the constant term to the right: Add '-0.6666666667' to each side of the equation. 0.6666666667 + -2y + -0.6666666667 + y2 = 0 + -0.6666666667 Reorder the terms: 0.6666666667 + -0.6666666667 + -2y + y2 = 0 + -0.6666666667 Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000 0.0000000000 + -2y + y2 = 0 + -0.6666666667 -2y + y2 = 0 + -0.6666666667 Combine like terms: 0 + -0.6666666667 = -0.6666666667 -2y + y2 = -0.6666666667 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 = -0.6666666667 + 1 Reorder the terms: 1 + -2y + y2 = -0.6666666667 + 1 Combine like terms: -0.6666666667 + 1 = 0.3333333333 1 + -2y + y2 = 0.3333333333 Factor a perfect square on the left side: (y + -1)(y + -1) = 0.3333333333 Calculate the square root of the right side: 0.577350269 Break this problem into two subproblems by setting (y + -1) equal to 0.577350269 and -0.577350269.Subproblem 1
y + -1 = 0.577350269 Simplifying y + -1 = 0.577350269 Reorder the terms: -1 + y = 0.577350269 Solving -1 + y = 0.577350269 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 = 0.577350269 + 1 Combine like terms: -1 + 1 = 0 0 + y = 0.577350269 + 1 y = 0.577350269 + 1 Combine like terms: 0.577350269 + 1 = 1.577350269 y = 1.577350269 Simplifying y = 1.577350269Subproblem 2
y + -1 = -0.577350269 Simplifying y + -1 = -0.577350269 Reorder the terms: -1 + y = -0.577350269 Solving -1 + y = -0.577350269 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 = -0.577350269 + 1 Combine like terms: -1 + 1 = 0 0 + y = -0.577350269 + 1 y = -0.577350269 + 1 Combine like terms: -0.577350269 + 1 = 0.422649731 y = 0.422649731 Simplifying y = 0.422649731Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.577350269, 0.422649731}Solution
y = {0, 1.577350269, 0.422649731}
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