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Simplifying 18y4 + 34y3 + -6y2 = 0 Reorder the terms: -6y2 + 34y3 + 18y4 = 0 Solving -6y2 + 34y3 + 18y4 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '2y2'. 2y2(-3 + 17y + 9y2) = 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 '(-3 + 17y + 9y2)' equal to zero and attempt to solve: Simplifying -3 + 17y + 9y2 = 0 Solving -3 + 17y + 9y2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.3333333333 + 1.888888889y + y2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + 1.888888889y + 0.3333333333 + y2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + 1.888888889y + y2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + 1.888888889y + y2 = 0 + 0.3333333333 1.888888889y + y2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 1.888888889y + y2 = 0.3333333333 The y term is 1.888888889y. Take half its coefficient (0.9444444445). Square it (0.8919753087) and add it to both sides. Add '0.8919753087' to each side of the equation. 1.888888889y + 0.8919753087 + y2 = 0.3333333333 + 0.8919753087 Reorder the terms: 0.8919753087 + 1.888888889y + y2 = 0.3333333333 + 0.8919753087 Combine like terms: 0.3333333333 + 0.8919753087 = 1.225308642 0.8919753087 + 1.888888889y + y2 = 1.225308642 Factor a perfect square on the left side: (y + 0.9444444445)(y + 0.9444444445) = 1.225308642 Calculate the square root of the right side: 1.106936603 Break this problem into two subproblems by setting (y + 0.9444444445) equal to 1.106936603 and -1.106936603.Subproblem 1
y + 0.9444444445 = 1.106936603 Simplifying y + 0.9444444445 = 1.106936603 Reorder the terms: 0.9444444445 + y = 1.106936603 Solving 0.9444444445 + y = 1.106936603 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.9444444445' to each side of the equation. 0.9444444445 + -0.9444444445 + y = 1.106936603 + -0.9444444445 Combine like terms: 0.9444444445 + -0.9444444445 = 0.0000000000 0.0000000000 + y = 1.106936603 + -0.9444444445 y = 1.106936603 + -0.9444444445 Combine like terms: 1.106936603 + -0.9444444445 = 0.1624921585 y = 0.1624921585 Simplifying y = 0.1624921585Subproblem 2
y + 0.9444444445 = -1.106936603 Simplifying y + 0.9444444445 = -1.106936603 Reorder the terms: 0.9444444445 + y = -1.106936603 Solving 0.9444444445 + y = -1.106936603 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.9444444445' to each side of the equation. 0.9444444445 + -0.9444444445 + y = -1.106936603 + -0.9444444445 Combine like terms: 0.9444444445 + -0.9444444445 = 0.0000000000 0.0000000000 + y = -1.106936603 + -0.9444444445 y = -1.106936603 + -0.9444444445 Combine like terms: -1.106936603 + -0.9444444445 = -2.0513810475 y = -2.0513810475 Simplifying y = -2.0513810475Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.1624921585, -2.0513810475}Solution
y = {0, 0.1624921585, -2.0513810475}
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