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Simplifying (3y + 5)(3y + 5) + (y + -4)(y + -4) = 6y(6y) Reorder the terms: (5 + 3y)(3y + 5) + (y + -4)(y + -4) = 6y(6y) Reorder the terms: (5 + 3y)(5 + 3y) + (y + -4)(y + -4) = 6y(6y) Multiply (5 + 3y) * (5 + 3y) (5(5 + 3y) + 3y * (5 + 3y)) + (y + -4)(y + -4) = 6y(6y) ((5 * 5 + 3y * 5) + 3y * (5 + 3y)) + (y + -4)(y + -4) = 6y(6y) ((25 + 15y) + 3y * (5 + 3y)) + (y + -4)(y + -4) = 6y(6y) (25 + 15y + (5 * 3y + 3y * 3y)) + (y + -4)(y + -4) = 6y(6y) (25 + 15y + (15y + 9y2)) + (y + -4)(y + -4) = 6y(6y) Combine like terms: 15y + 15y = 30y (25 + 30y + 9y2) + (y + -4)(y + -4) = 6y(6y) Reorder the terms: 25 + 30y + 9y2 + (-4 + y)(y + -4) = 6y(6y) Reorder the terms: 25 + 30y + 9y2 + (-4 + y)(-4 + y) = 6y(6y) Multiply (-4 + y) * (-4 + y) 25 + 30y + 9y2 + (-4(-4 + y) + y(-4 + y)) = 6y(6y) 25 + 30y + 9y2 + ((-4 * -4 + y * -4) + y(-4 + y)) = 6y(6y) 25 + 30y + 9y2 + ((16 + -4y) + y(-4 + y)) = 6y(6y) 25 + 30y + 9y2 + (16 + -4y + (-4 * y + y * y)) = 6y(6y) 25 + 30y + 9y2 + (16 + -4y + (-4y + y2)) = 6y(6y) Combine like terms: -4y + -4y = -8y 25 + 30y + 9y2 + (16 + -8y + y2) = 6y(6y) Reorder the terms: 25 + 16 + 30y + -8y + 9y2 + y2 = 6y(6y) Combine like terms: 25 + 16 = 41 41 + 30y + -8y + 9y2 + y2 = 6y(6y) Combine like terms: 30y + -8y = 22y 41 + 22y + 9y2 + y2 = 6y(6y) Combine like terms: 9y2 + y2 = 10y2 41 + 22y + 10y2 = 6y(6y) Remove parenthesis around (6y) 41 + 22y + 10y2 = 6y * 6y Reorder the terms for easier multiplication: 41 + 22y + 10y2 = 6 * 6y * y Multiply 6 * 6 41 + 22y + 10y2 = 36y * y Multiply y * y 41 + 22y + 10y2 = 36y2 Solving 41 + 22y + 10y2 = 36y2 Solving for variable 'y'. Combine like terms: 10y2 + -36y2 = -26y2 41 + 22y + -26y2 = 36y2 + -36y2 Combine like terms: 36y2 + -36y2 = 0 41 + 22y + -26y2 = 0 Begin completing the square. Divide all terms by -26 the coefficient of the squared term: Divide each side by '-26'. -1.576923077 + -0.8461538462y + y2 = 0 Move the constant term to the right: Add '1.576923077' to each side of the equation. -1.576923077 + -0.8461538462y + 1.576923077 + y2 = 0 + 1.576923077 Reorder the terms: -1.576923077 + 1.576923077 + -0.8461538462y + y2 = 0 + 1.576923077 Combine like terms: -1.576923077 + 1.576923077 = 0.000000000 0.000000000 + -0.8461538462y + y2 = 0 + 1.576923077 -0.8461538462y + y2 = 0 + 1.576923077 Combine like terms: 0 + 1.576923077 = 1.576923077 -0.8461538462y + y2 = 1.576923077 The y term is -0.8461538462y. Take half its coefficient (-0.4230769231). Square it (0.1789940829) and add it to both sides. Add '0.1789940829' to each side of the equation. -0.8461538462y + 0.1789940829 + y2 = 1.576923077 + 0.1789940829 Reorder the terms: 0.1789940829 + -0.8461538462y + y2 = 1.576923077 + 0.1789940829 Combine like terms: 1.576923077 + 0.1789940829 = 1.7559171599 0.1789940829 + -0.8461538462y + y2 = 1.7559171599 Factor a perfect square on the left side: (y + -0.4230769231)(y + -0.4230769231) = 1.7559171599 Calculate the square root of the right side: 1.325110244 Break this problem into two subproblems by setting (y + -0.4230769231) equal to 1.325110244 and -1.325110244.Subproblem 1
y + -0.4230769231 = 1.325110244 Simplifying y + -0.4230769231 = 1.325110244 Reorder the terms: -0.4230769231 + y = 1.325110244 Solving -0.4230769231 + y = 1.325110244 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.4230769231' to each side of the equation. -0.4230769231 + 0.4230769231 + y = 1.325110244 + 0.4230769231 Combine like terms: -0.4230769231 + 0.4230769231 = 0.0000000000 0.0000000000 + y = 1.325110244 + 0.4230769231 y = 1.325110244 + 0.4230769231 Combine like terms: 1.325110244 + 0.4230769231 = 1.7481871671 y = 1.7481871671 Simplifying y = 1.7481871671Subproblem 2
y + -0.4230769231 = -1.325110244 Simplifying y + -0.4230769231 = -1.325110244 Reorder the terms: -0.4230769231 + y = -1.325110244 Solving -0.4230769231 + y = -1.325110244 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.4230769231' to each side of the equation. -0.4230769231 + 0.4230769231 + y = -1.325110244 + 0.4230769231 Combine like terms: -0.4230769231 + 0.4230769231 = 0.0000000000 0.0000000000 + y = -1.325110244 + 0.4230769231 y = -1.325110244 + 0.4230769231 Combine like terms: -1.325110244 + 0.4230769231 = -0.9020333209 y = -0.9020333209 Simplifying y = -0.9020333209Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.7481871671, -0.9020333209}
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