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Simplifying 3y2 + -2y + 5 = 10y + 1 Reorder the terms: 5 + -2y + 3y2 = 10y + 1 Reorder the terms: 5 + -2y + 3y2 = 1 + 10y Solving 5 + -2y + 3y2 = 1 + 10y Solving for variable 'y'. Reorder the terms: 5 + -1 + -2y + -10y + 3y2 = 1 + 10y + -1 + -10y Combine like terms: 5 + -1 = 4 4 + -2y + -10y + 3y2 = 1 + 10y + -1 + -10y Combine like terms: -2y + -10y = -12y 4 + -12y + 3y2 = 1 + 10y + -1 + -10y Reorder the terms: 4 + -12y + 3y2 = 1 + -1 + 10y + -10y Combine like terms: 1 + -1 = 0 4 + -12y + 3y2 = 0 + 10y + -10y 4 + -12y + 3y2 = 10y + -10y Combine like terms: 10y + -10y = 0 4 + -12y + 3y2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1.333333333 + -4y + y2 = 0 Move the constant term to the right: Add '-1.333333333' to each side of the equation. 1.333333333 + -4y + -1.333333333 + y2 = 0 + -1.333333333 Reorder the terms: 1.333333333 + -1.333333333 + -4y + y2 = 0 + -1.333333333 Combine like terms: 1.333333333 + -1.333333333 = 0.000000000 0.000000000 + -4y + y2 = 0 + -1.333333333 -4y + y2 = 0 + -1.333333333 Combine like terms: 0 + -1.333333333 = -1.333333333 -4y + y2 = -1.333333333 The y term is -4y. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4y + 4 + y2 = -1.333333333 + 4 Reorder the terms: 4 + -4y + y2 = -1.333333333 + 4 Combine like terms: -1.333333333 + 4 = 2.666666667 4 + -4y + y2 = 2.666666667 Factor a perfect square on the left side: (y + -2)(y + -2) = 2.666666667 Calculate the square root of the right side: 1.632993162 Break this problem into two subproblems by setting (y + -2) equal to 1.632993162 and -1.632993162.Subproblem 1
y + -2 = 1.632993162 Simplifying y + -2 = 1.632993162 Reorder the terms: -2 + y = 1.632993162 Solving -2 + y = 1.632993162 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + y = 1.632993162 + 2 Combine like terms: -2 + 2 = 0 0 + y = 1.632993162 + 2 y = 1.632993162 + 2 Combine like terms: 1.632993162 + 2 = 3.632993162 y = 3.632993162 Simplifying y = 3.632993162Subproblem 2
y + -2 = -1.632993162 Simplifying y + -2 = -1.632993162 Reorder the terms: -2 + y = -1.632993162 Solving -2 + y = -1.632993162 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + y = -1.632993162 + 2 Combine like terms: -2 + 2 = 0 0 + y = -1.632993162 + 2 y = -1.632993162 + 2 Combine like terms: -1.632993162 + 2 = 0.367006838 y = 0.367006838 Simplifying y = 0.367006838Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.632993162, 0.367006838}
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