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Simplifying (27y2) + -12y + -3 = 0 Reorder the terms: -3 + -12y + (27y2) = 0 Solving -3 + -12y + (27y2) = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '3'. 3(-1 + -4y + (9y2)) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-1 + -4y + (9y2))' equal to zero and attempt to solve: Simplifying -1 + -4y + (9y2) = 0 Solving -1 + -4y + (9y2) = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.1111111111 + -0.4444444444y + y2 = 0 Move the constant term to the right: Add '0.1111111111' to each side of the equation. -0.1111111111 + -0.4444444444y + 0.1111111111 + y2 = 0 + 0.1111111111 Reorder the terms: -0.1111111111 + 0.1111111111 + -0.4444444444y + y2 = 0 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + -0.4444444444y + y2 = 0 + 0.1111111111 -0.4444444444y + y2 = 0 + 0.1111111111 Combine like terms: 0 + 0.1111111111 = 0.1111111111 -0.4444444444y + y2 = 0.1111111111 The y term is -0.4444444444y. Take half its coefficient (-0.2222222222). Square it (0.04938271604) and add it to both sides. Add '0.04938271604' to each side of the equation. -0.4444444444y + 0.04938271604 + y2 = 0.1111111111 + 0.04938271604 Reorder the terms: 0.04938271604 + -0.4444444444y + y2 = 0.1111111111 + 0.04938271604 Combine like terms: 0.1111111111 + 0.04938271604 = 0.16049382714 0.04938271604 + -0.4444444444y + y2 = 0.16049382714 Factor a perfect square on the left side: ((y) + -0.2222222222)((y) + -0.2222222222) = 0.16049382714 Calculate the square root of the right side: 0.400616808 Break this problem into two subproblems by setting ((y) + -0.2222222222) equal to 0.400616808 and -0.400616808.Subproblem 1
(y) + -0.2222222222 = 0.400616808 Simplifying (y) + -0.2222222222 = 0.400616808 y + -0.2222222222 = 0.400616808 Reorder the terms: -0.2222222222 + y = 0.400616808 Solving -0.2222222222 + y = 0.400616808 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + y = 0.400616808 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + y = 0.400616808 + 0.2222222222 y = 0.400616808 + 0.2222222222 Combine like terms: 0.400616808 + 0.2222222222 = 0.6228390302 y = 0.6228390302 Simplifying y = 0.6228390302Subproblem 2
(y) + -0.2222222222 = -0.400616808 Simplifying (y) + -0.2222222222 = -0.400616808 y + -0.2222222222 = -0.400616808 Reorder the terms: -0.2222222222 + y = -0.400616808 Solving -0.2222222222 + y = -0.400616808 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + y = -0.400616808 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + y = -0.400616808 + 0.2222222222 y = -0.400616808 + 0.2222222222 Combine like terms: -0.400616808 + 0.2222222222 = -0.1783945858 y = -0.1783945858 Simplifying y = -0.1783945858Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.6228390302, -0.1783945858}Solution
y = {0.6228390302, -0.1783945858}
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