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Simplifying (5y + 2y2 + 3) + (-5y2 + 2y) = 0 Reorder the terms: (3 + 5y + 2y2) + (-5y2 + 2y) = 0 Remove parenthesis around (3 + 5y + 2y2) 3 + 5y + 2y2 + (-5y2 + 2y) = 0 Reorder the terms: 3 + 5y + 2y2 + (2y + -5y2) = 0 Remove parenthesis around (2y + -5y2) 3 + 5y + 2y2 + 2y + -5y2 = 0 Reorder the terms: 3 + 5y + 2y + 2y2 + -5y2 = 0 Combine like terms: 5y + 2y = 7y 3 + 7y + 2y2 + -5y2 = 0 Combine like terms: 2y2 + -5y2 = -3y2 3 + 7y + -3y2 = 0 Solving 3 + 7y + -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 + -2.333333333y + y2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -2.333333333y + 1 + y2 = 0 + 1 Reorder the terms: -1 + 1 + -2.333333333y + y2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -2.333333333y + y2 = 0 + 1 -2.333333333y + y2 = 0 + 1 Combine like terms: 0 + 1 = 1 -2.333333333y + y2 = 1 The y term is -2.333333333y. Take half its coefficient (-1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. -2.333333333y + 1.361111112 + y2 = 1 + 1.361111112 Reorder the terms: 1.361111112 + -2.333333333y + y2 = 1 + 1.361111112 Combine like terms: 1 + 1.361111112 = 2.361111112 1.361111112 + -2.333333333y + y2 = 2.361111112 Factor a perfect square on the left side: (y + -1.166666667)(y + -1.166666667) = 2.361111112 Calculate the square root of the right side: 1.536590743 Break this problem into two subproblems by setting (y + -1.166666667) equal to 1.536590743 and -1.536590743.Subproblem 1
y + -1.166666667 = 1.536590743 Simplifying y + -1.166666667 = 1.536590743 Reorder the terms: -1.166666667 + y = 1.536590743 Solving -1.166666667 + y = 1.536590743 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + y = 1.536590743 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + y = 1.536590743 + 1.166666667 y = 1.536590743 + 1.166666667 Combine like terms: 1.536590743 + 1.166666667 = 2.70325741 y = 2.70325741 Simplifying y = 2.70325741Subproblem 2
y + -1.166666667 = -1.536590743 Simplifying y + -1.166666667 = -1.536590743 Reorder the terms: -1.166666667 + y = -1.536590743 Solving -1.166666667 + y = -1.536590743 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + y = -1.536590743 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + y = -1.536590743 + 1.166666667 y = -1.536590743 + 1.166666667 Combine like terms: -1.536590743 + 1.166666667 = -0.369924076 y = -0.369924076 Simplifying y = -0.369924076Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.70325741, -0.369924076}
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