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Simplifying 3y2 + 9y + 1 = 0 Reorder the terms: 1 + 9y + 3y2 = 0 Solving 1 + 9y + 3y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.3333333333 + 3y + y2 = 0 Move the constant term to the right: Add '-0.3333333333' to each side of the equation. 0.3333333333 + 3y + -0.3333333333 + y2 = 0 + -0.3333333333 Reorder the terms: 0.3333333333 + -0.3333333333 + 3y + y2 = 0 + -0.3333333333 Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000 0.0000000000 + 3y + y2 = 0 + -0.3333333333 3y + y2 = 0 + -0.3333333333 Combine like terms: 0 + -0.3333333333 = -0.3333333333 3y + y2 = -0.3333333333 The y term is 3y. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3y + 2.25 + y2 = -0.3333333333 + 2.25 Reorder the terms: 2.25 + 3y + y2 = -0.3333333333 + 2.25 Combine like terms: -0.3333333333 + 2.25 = 1.9166666667 2.25 + 3y + y2 = 1.9166666667 Factor a perfect square on the left side: (y + 1.5)(y + 1.5) = 1.9166666667 Calculate the square root of the right side: 1.38443731 Break this problem into two subproblems by setting (y + 1.5) equal to 1.38443731 and -1.38443731.Subproblem 1
y + 1.5 = 1.38443731 Simplifying y + 1.5 = 1.38443731 Reorder the terms: 1.5 + y = 1.38443731 Solving 1.5 + y = 1.38443731 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = 1.38443731 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = 1.38443731 + -1.5 y = 1.38443731 + -1.5 Combine like terms: 1.38443731 + -1.5 = -0.11556269 y = -0.11556269 Simplifying y = -0.11556269Subproblem 2
y + 1.5 = -1.38443731 Simplifying y + 1.5 = -1.38443731 Reorder the terms: 1.5 + y = -1.38443731 Solving 1.5 + y = -1.38443731 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = -1.38443731 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = -1.38443731 + -1.5 y = -1.38443731 + -1.5 Combine like terms: -1.38443731 + -1.5 = -2.88443731 y = -2.88443731 Simplifying y = -2.88443731Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.11556269, -2.88443731}
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