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Simplifying (3y2 + -5y) + (6y2 + y + -9) = 0 Reorder the terms: (-5y + 3y2) + (6y2 + y + -9) = 0 Remove parenthesis around (-5y + 3y2) -5y + 3y2 + (6y2 + y + -9) = 0 Reorder the terms: -5y + 3y2 + (-9 + y + 6y2) = 0 Remove parenthesis around (-9 + y + 6y2) -5y + 3y2 + -9 + y + 6y2 = 0 Reorder the terms: -9 + -5y + y + 3y2 + 6y2 = 0 Combine like terms: -5y + y = -4y -9 + -4y + 3y2 + 6y2 = 0 Combine like terms: 3y2 + 6y2 = 9y2 -9 + -4y + 9y2 = 0 Solving -9 + -4y + 9y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1 + -0.4444444444y + y2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.4444444444y + 1 + y2 = 0 + 1 Reorder the terms: -1 + 1 + -0.4444444444y + y2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.4444444444y + y2 = 0 + 1 -0.4444444444y + y2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.4444444444y + y2 = 1 The y term is -0.4444444444y. Take half its coefficient (-0.2222222222). Square it (0.04938271604) and add it to both sides. Add '0.04938271604' to each side of the equation. -0.4444444444y + 0.04938271604 + y2 = 1 + 0.04938271604 Reorder the terms: 0.04938271604 + -0.4444444444y + y2 = 1 + 0.04938271604 Combine like terms: 1 + 0.04938271604 = 1.04938271604 0.04938271604 + -0.4444444444y + y2 = 1.04938271604 Factor a perfect square on the left side: (y + -0.2222222222)(y + -0.2222222222) = 1.04938271604 Calculate the square root of the right side: 1.024393829 Break this problem into two subproblems by setting (y + -0.2222222222) equal to 1.024393829 and -1.024393829.Subproblem 1
y + -0.2222222222 = 1.024393829 Simplifying y + -0.2222222222 = 1.024393829 Reorder the terms: -0.2222222222 + y = 1.024393829 Solving -0.2222222222 + y = 1.024393829 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + y = 1.024393829 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + y = 1.024393829 + 0.2222222222 y = 1.024393829 + 0.2222222222 Combine like terms: 1.024393829 + 0.2222222222 = 1.2466160512 y = 1.2466160512 Simplifying y = 1.2466160512Subproblem 2
y + -0.2222222222 = -1.024393829 Simplifying y + -0.2222222222 = -1.024393829 Reorder the terms: -0.2222222222 + y = -1.024393829 Solving -0.2222222222 + y = -1.024393829 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + y = -1.024393829 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + y = -1.024393829 + 0.2222222222 y = -1.024393829 + 0.2222222222 Combine like terms: -1.024393829 + 0.2222222222 = -0.8021716068 y = -0.8021716068 Simplifying y = -0.8021716068Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.2466160512, -0.8021716068}
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