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Simplifying 6 + -2y + -6(-3y + 7) * y = 3 Reorder the terms: 6 + -2y + -6(7 + -3y) * y = 3 Reorder the terms for easier multiplication: 6 + -2y + -6y(7 + -3y) = 3 6 + -2y + (7 * -6y + -3y * -6y) = 3 6 + -2y + (-42y + 18y2) = 3 Combine like terms: -2y + -42y = -44y 6 + -44y + 18y2 = 3 Solving 6 + -44y + 18y2 = 3 Solving for variable 'y'. Reorder the terms: 6 + -3 + -44y + 18y2 = 3 + -3 Combine like terms: 6 + -3 = 3 3 + -44y + 18y2 = 3 + -3 Combine like terms: 3 + -3 = 0 3 + -44y + 18y2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. 0.1666666667 + -2.444444444y + y2 = 0 Move the constant term to the right: Add '-0.1666666667' to each side of the equation. 0.1666666667 + -2.444444444y + -0.1666666667 + y2 = 0 + -0.1666666667 Reorder the terms: 0.1666666667 + -0.1666666667 + -2.444444444y + y2 = 0 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + -2.444444444y + y2 = 0 + -0.1666666667 -2.444444444y + y2 = 0 + -0.1666666667 Combine like terms: 0 + -0.1666666667 = -0.1666666667 -2.444444444y + y2 = -0.1666666667 The y term is -2.444444444y. Take half its coefficient (-1.222222222). Square it (1.493827160) and add it to both sides. Add '1.493827160' to each side of the equation. -2.444444444y + 1.493827160 + y2 = -0.1666666667 + 1.493827160 Reorder the terms: 1.493827160 + -2.444444444y + y2 = -0.1666666667 + 1.493827160 Combine like terms: -0.1666666667 + 1.493827160 = 1.3271604933 1.493827160 + -2.444444444y + y2 = 1.3271604933 Factor a perfect square on the left side: (y + -1.222222222)(y + -1.222222222) = 1.3271604933 Calculate the square root of the right side: 1.152024519 Break this problem into two subproblems by setting (y + -1.222222222) equal to 1.152024519 and -1.152024519.Subproblem 1
y + -1.222222222 = 1.152024519 Simplifying y + -1.222222222 = 1.152024519 Reorder the terms: -1.222222222 + y = 1.152024519 Solving -1.222222222 + y = 1.152024519 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.222222222' to each side of the equation. -1.222222222 + 1.222222222 + y = 1.152024519 + 1.222222222 Combine like terms: -1.222222222 + 1.222222222 = 0.000000000 0.000000000 + y = 1.152024519 + 1.222222222 y = 1.152024519 + 1.222222222 Combine like terms: 1.152024519 + 1.222222222 = 2.374246741 y = 2.374246741 Simplifying y = 2.374246741Subproblem 2
y + -1.222222222 = -1.152024519 Simplifying y + -1.222222222 = -1.152024519 Reorder the terms: -1.222222222 + y = -1.152024519 Solving -1.222222222 + y = -1.152024519 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.222222222' to each side of the equation. -1.222222222 + 1.222222222 + y = -1.152024519 + 1.222222222 Combine like terms: -1.222222222 + 1.222222222 = 0.000000000 0.000000000 + y = -1.152024519 + 1.222222222 y = -1.152024519 + 1.222222222 Combine like terms: -1.152024519 + 1.222222222 = 0.070197703 y = 0.070197703 Simplifying y = 0.070197703Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.374246741, 0.070197703}
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