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Simplifying 0 = 6y4 + -8y3 + -2y2 Reorder the terms: 0 = -2y2 + -8y3 + 6y4 Solving 0 = -2y2 + -8y3 + 6y4 Solving for variable 'y'. Remove the zero: 2y2 + 8y3 + -6y4 = -2y2 + -8y3 + 6y4 + 2y2 + 8y3 + -6y4 Reorder the terms: 2y2 + 8y3 + -6y4 = -2y2 + 2y2 + -8y3 + 8y3 + 6y4 + -6y4 Combine like terms: -2y2 + 2y2 = 0 2y2 + 8y3 + -6y4 = 0 + -8y3 + 8y3 + 6y4 + -6y4 2y2 + 8y3 + -6y4 = -8y3 + 8y3 + 6y4 + -6y4 Combine like terms: -8y3 + 8y3 = 0 2y2 + 8y3 + -6y4 = 0 + 6y4 + -6y4 2y2 + 8y3 + -6y4 = 6y4 + -6y4 Combine like terms: 6y4 + -6y4 = 0 2y2 + 8y3 + -6y4 = 0 Factor out the Greatest Common Factor (GCF), '2y2'. 2y2(1 + 4y + -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 '(1 + 4y + -3y2)' equal to zero and attempt to solve: Simplifying 1 + 4y + -3y2 = 0 Solving 1 + 4y + -3y2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -0.3333333333 + -1.333333333y + y2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + -1.333333333y + 0.3333333333 + y2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + -1.333333333y + y2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + -1.333333333y + y2 = 0 + 0.3333333333 -1.333333333y + y2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 -1.333333333y + y2 = 0.3333333333 The y term is -1.333333333y. Take half its coefficient (-0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. -1.333333333y + 0.4444444442 + y2 = 0.3333333333 + 0.4444444442 Reorder the terms: 0.4444444442 + -1.333333333y + y2 = 0.3333333333 + 0.4444444442 Combine like terms: 0.3333333333 + 0.4444444442 = 0.7777777775 0.4444444442 + -1.333333333y + y2 = 0.7777777775 Factor a perfect square on the left side: (y + -0.6666666665)(y + -0.6666666665) = 0.7777777775 Calculate the square root of the right side: 0.881917104 Break this problem into two subproblems by setting (y + -0.6666666665) equal to 0.881917104 and -0.881917104.Subproblem 1
y + -0.6666666665 = 0.881917104 Simplifying y + -0.6666666665 = 0.881917104 Reorder the terms: -0.6666666665 + y = 0.881917104 Solving -0.6666666665 + y = 0.881917104 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + y = 0.881917104 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + y = 0.881917104 + 0.6666666665 y = 0.881917104 + 0.6666666665 Combine like terms: 0.881917104 + 0.6666666665 = 1.5485837705 y = 1.5485837705 Simplifying y = 1.5485837705Subproblem 2
y + -0.6666666665 = -0.881917104 Simplifying y + -0.6666666665 = -0.881917104 Reorder the terms: -0.6666666665 + y = -0.881917104 Solving -0.6666666665 + y = -0.881917104 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + y = -0.881917104 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + y = -0.881917104 + 0.6666666665 y = -0.881917104 + 0.6666666665 Combine like terms: -0.881917104 + 0.6666666665 = -0.2152504375 y = -0.2152504375 Simplifying y = -0.2152504375Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.5485837705, -0.2152504375}Solution
y = {0, 1.5485837705, -0.2152504375}
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