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Simplifying 3y2 + [6y2 + (3y + -4)] = 0 Reorder the terms: 3y2 + [6y2 + (-4 + 3y)] = 0 Remove parenthesis around (-4 + 3y) 3y2 + [6y2 + -4 + 3y] = 0 Reorder the terms: 3y2 + [-4 + 3y + 6y2] = 0 Remove brackets around [-4 + 3y + 6y2] 3y2 + -4 + 3y + 6y2 = 0 Reorder the terms: -4 + 3y + 3y2 + 6y2 = 0 Combine like terms: 3y2 + 6y2 = 9y2 -4 + 3y + 9y2 = 0 Solving -4 + 3y + 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'. -0.4444444444 + 0.3333333333y + y2 = 0 Move the constant term to the right: Add '0.4444444444' to each side of the equation. -0.4444444444 + 0.3333333333y + 0.4444444444 + y2 = 0 + 0.4444444444 Reorder the terms: -0.4444444444 + 0.4444444444 + 0.3333333333y + y2 = 0 + 0.4444444444 Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000 0.0000000000 + 0.3333333333y + y2 = 0 + 0.4444444444 0.3333333333y + y2 = 0 + 0.4444444444 Combine like terms: 0 + 0.4444444444 = 0.4444444444 0.3333333333y + y2 = 0.4444444444 The y term is 0.3333333333y. Take half its coefficient (0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. 0.3333333333y + 0.02777777779 + y2 = 0.4444444444 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333y + y2 = 0.4444444444 + 0.02777777779 Combine like terms: 0.4444444444 + 0.02777777779 = 0.47222222219 0.02777777779 + 0.3333333333y + y2 = 0.47222222219 Factor a perfect square on the left side: (y + 0.1666666667)(y + 0.1666666667) = 0.47222222219 Calculate the square root of the right side: 0.687184271 Break this problem into two subproblems by setting (y + 0.1666666667) equal to 0.687184271 and -0.687184271.Subproblem 1
y + 0.1666666667 = 0.687184271 Simplifying y + 0.1666666667 = 0.687184271 Reorder the terms: 0.1666666667 + y = 0.687184271 Solving 0.1666666667 + y = 0.687184271 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + y = 0.687184271 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + y = 0.687184271 + -0.1666666667 y = 0.687184271 + -0.1666666667 Combine like terms: 0.687184271 + -0.1666666667 = 0.5205176043 y = 0.5205176043 Simplifying y = 0.5205176043Subproblem 2
y + 0.1666666667 = -0.687184271 Simplifying y + 0.1666666667 = -0.687184271 Reorder the terms: 0.1666666667 + y = -0.687184271 Solving 0.1666666667 + y = -0.687184271 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + y = -0.687184271 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + y = -0.687184271 + -0.1666666667 y = -0.687184271 + -0.1666666667 Combine like terms: -0.687184271 + -0.1666666667 = -0.8538509377 y = -0.8538509377 Simplifying y = -0.8538509377Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.5205176043, -0.8538509377}
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