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Simplifying 3y2 + -18y + 19 = 0 Reorder the terms: 19 + -18y + 3y2 = 0 Solving 19 + -18y + 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'. 6.333333333 + -6y + y2 = 0 Move the constant term to the right: Add '-6.333333333' to each side of the equation. 6.333333333 + -6y + -6.333333333 + y2 = 0 + -6.333333333 Reorder the terms: 6.333333333 + -6.333333333 + -6y + y2 = 0 + -6.333333333 Combine like terms: 6.333333333 + -6.333333333 = 0.000000000 0.000000000 + -6y + y2 = 0 + -6.333333333 -6y + y2 = 0 + -6.333333333 Combine like terms: 0 + -6.333333333 = -6.333333333 -6y + y2 = -6.333333333 The y term is -6y. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6y + 9 + y2 = -6.333333333 + 9 Reorder the terms: 9 + -6y + y2 = -6.333333333 + 9 Combine like terms: -6.333333333 + 9 = 2.666666667 9 + -6y + y2 = 2.666666667 Factor a perfect square on the left side: (y + -3)(y + -3) = 2.666666667 Calculate the square root of the right side: 1.632993162 Break this problem into two subproblems by setting (y + -3) equal to 1.632993162 and -1.632993162.Subproblem 1
y + -3 = 1.632993162 Simplifying y + -3 = 1.632993162 Reorder the terms: -3 + y = 1.632993162 Solving -3 + y = 1.632993162 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + y = 1.632993162 + 3 Combine like terms: -3 + 3 = 0 0 + y = 1.632993162 + 3 y = 1.632993162 + 3 Combine like terms: 1.632993162 + 3 = 4.632993162 y = 4.632993162 Simplifying y = 4.632993162Subproblem 2
y + -3 = -1.632993162 Simplifying y + -3 = -1.632993162 Reorder the terms: -3 + y = -1.632993162 Solving -3 + y = -1.632993162 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + y = -1.632993162 + 3 Combine like terms: -3 + 3 = 0 0 + y = -1.632993162 + 3 y = -1.632993162 + 3 Combine like terms: -1.632993162 + 3 = 1.367006838 y = 1.367006838 Simplifying y = 1.367006838Solution
The solution to the problem is based on the solutions from the subproblems. y = {4.632993162, 1.367006838}
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