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Simplifying 3y2 + -9y + 5 = 0 Reorder the terms: 5 + -9y + 3y2 = 0 Solving 5 + -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'. 1.666666667 + -3y + y2 = 0 Move the constant term to the right: Add '-1.666666667' to each side of the equation. 1.666666667 + -3y + -1.666666667 + y2 = 0 + -1.666666667 Reorder the terms: 1.666666667 + -1.666666667 + -3y + y2 = 0 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + -3y + y2 = 0 + -1.666666667 -3y + y2 = 0 + -1.666666667 Combine like terms: 0 + -1.666666667 = -1.666666667 -3y + y2 = -1.666666667 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 = -1.666666667 + 2.25 Reorder the terms: 2.25 + -3y + y2 = -1.666666667 + 2.25 Combine like terms: -1.666666667 + 2.25 = 0.583333333 2.25 + -3y + y2 = 0.583333333 Factor a perfect square on the left side: (y + -1.5)(y + -1.5) = 0.583333333 Calculate the square root of the right side: 0.763762616 Break this problem into two subproblems by setting (y + -1.5) equal to 0.763762616 and -0.763762616.Subproblem 1
y + -1.5 = 0.763762616 Simplifying y + -1.5 = 0.763762616 Reorder the terms: -1.5 + y = 0.763762616 Solving -1.5 + y = 0.763762616 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 = 0.763762616 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = 0.763762616 + 1.5 y = 0.763762616 + 1.5 Combine like terms: 0.763762616 + 1.5 = 2.263762616 y = 2.263762616 Simplifying y = 2.263762616Subproblem 2
y + -1.5 = -0.763762616 Simplifying y + -1.5 = -0.763762616 Reorder the terms: -1.5 + y = -0.763762616 Solving -1.5 + y = -0.763762616 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 = -0.763762616 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = -0.763762616 + 1.5 y = -0.763762616 + 1.5 Combine like terms: -0.763762616 + 1.5 = 0.736237384 y = 0.736237384 Simplifying y = 0.736237384Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.263762616, 0.736237384}
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