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Simplifying 25 + -1((2 + 5y + -3)(y + 2)) = -3(2y + -5) + -1(5(y + -1) + -3y + 3) Reorder the terms: 25 + -1((2 + -3 + 5y)(y + 2)) = -3(2y + -5) + -1(5(y + -1) + -3y + 3) Combine like terms: 2 + -3 = -1 25 + -1((-1 + 5y)(y + 2)) = -3(2y + -5) + -1(5(y + -1) + -3y + 3) Reorder the terms: 25 + -1((-1 + 5y)(2 + y)) = -3(2y + -5) + -1(5(y + -1) + -3y + 3) Multiply (-1 + 5y) * (2 + y) 25 + -1((-1(2 + y) + 5y * (2 + y))) = -3(2y + -5) + -1(5(y + -1) + -3y + 3) 25 + -1(((2 * -1 + y * -1) + 5y * (2 + y))) = -3(2y + -5) + -1(5(y + -1) + -3y + 3) 25 + -1(((-2 + -1y) + 5y * (2 + y))) = -3(2y + -5) + -1(5(y + -1) + -3y + 3) 25 + -1((-2 + -1y + (2 * 5y + y * 5y))) = -3(2y + -5) + -1(5(y + -1) + -3y + 3) 25 + -1((-2 + -1y + (10y + 5y2))) = -3(2y + -5) + -1(5(y + -1) + -3y + 3) Combine like terms: -1y + 10y = 9y 25 + -1((-2 + 9y + 5y2)) = -3(2y + -5) + -1(5(y + -1) + -3y + 3) 25 + (-2 * -1 + 9y * -1 + 5y2 * -1) = -3(2y + -5) + -1(5(y + -1) + -3y + 3) 25 + (2 + -9y + -5y2) = -3(2y + -5) + -1(5(y + -1) + -3y + 3) Combine like terms: 25 + 2 = 27 27 + -9y + -5y2 = -3(2y + -5) + -1(5(y + -1) + -3y + 3) Reorder the terms: 27 + -9y + -5y2 = -3(-5 + 2y) + -1(5(y + -1) + -3y + 3) 27 + -9y + -5y2 = (-5 * -3 + 2y * -3) + -1(5(y + -1) + -3y + 3) 27 + -9y + -5y2 = (15 + -6y) + -1(5(y + -1) + -3y + 3) Reorder the terms: 27 + -9y + -5y2 = 15 + -6y + -1(5(-1 + y) + -3y + 3) 27 + -9y + -5y2 = 15 + -6y + -1((-1 * 5 + y * 5) + -3y + 3) 27 + -9y + -5y2 = 15 + -6y + -1((-5 + 5y) + -3y + 3) Reorder the terms: 27 + -9y + -5y2 = 15 + -6y + -1(-5 + 3 + 5y + -3y) Combine like terms: -5 + 3 = -2 27 + -9y + -5y2 = 15 + -6y + -1(-2 + 5y + -3y) Combine like terms: 5y + -3y = 2y 27 + -9y + -5y2 = 15 + -6y + -1(-2 + 2y) 27 + -9y + -5y2 = 15 + -6y + (-2 * -1 + 2y * -1) 27 + -9y + -5y2 = 15 + -6y + (2 + -2y) Reorder the terms: 27 + -9y + -5y2 = 15 + 2 + -6y + -2y Combine like terms: 15 + 2 = 17 27 + -9y + -5y2 = 17 + -6y + -2y Combine like terms: -6y + -2y = -8y 27 + -9y + -5y2 = 17 + -8y Solving 27 + -9y + -5y2 = 17 + -8y Solving for variable 'y'. Reorder the terms: 27 + -17 + -9y + 8y + -5y2 = 17 + -8y + -17 + 8y Combine like terms: 27 + -17 = 10 10 + -9y + 8y + -5y2 = 17 + -8y + -17 + 8y Combine like terms: -9y + 8y = -1y 10 + -1y + -5y2 = 17 + -8y + -17 + 8y Reorder the terms: 10 + -1y + -5y2 = 17 + -17 + -8y + 8y Combine like terms: 17 + -17 = 0 10 + -1y + -5y2 = 0 + -8y + 8y 10 + -1y + -5y2 = -8y + 8y Combine like terms: -8y + 8y = 0 10 + -1y + -5y2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -2 + 0.2y + y2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 0.2y + 2 + y2 = 0 + 2 Reorder the terms: -2 + 2 + 0.2y + y2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 0.2y + y2 = 0 + 2 0.2y + y2 = 0 + 2 Combine like terms: 0 + 2 = 2 0.2y + y2 = 2 The y term is 0.2y. Take half its coefficient (0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. 0.2y + 0.01 + y2 = 2 + 0.01 Reorder the terms: 0.01 + 0.2y + y2 = 2 + 0.01 Combine like terms: 2 + 0.01 = 2.01 0.01 + 0.2y + y2 = 2.01 Factor a perfect square on the left side: (y + 0.1)(y + 0.1) = 2.01 Calculate the square root of the right side: 1.417744688 Break this problem into two subproblems by setting (y + 0.1) equal to 1.417744688 and -1.417744688.Subproblem 1
y + 0.1 = 1.417744688 Simplifying y + 0.1 = 1.417744688 Reorder the terms: 0.1 + y = 1.417744688 Solving 0.1 + y = 1.417744688 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.1' to each side of the equation. 0.1 + -0.1 + y = 1.417744688 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + y = 1.417744688 + -0.1 y = 1.417744688 + -0.1 Combine like terms: 1.417744688 + -0.1 = 1.317744688 y = 1.317744688 Simplifying y = 1.317744688Subproblem 2
y + 0.1 = -1.417744688 Simplifying y + 0.1 = -1.417744688 Reorder the terms: 0.1 + y = -1.417744688 Solving 0.1 + y = -1.417744688 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.1' to each side of the equation. 0.1 + -0.1 + y = -1.417744688 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + y = -1.417744688 + -0.1 y = -1.417744688 + -0.1 Combine like terms: -1.417744688 + -0.1 = -1.517744688 y = -1.517744688 Simplifying y = -1.517744688Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.317744688, -1.517744688}
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