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Simplifying (5 + -5y) + -20 = -1(5y + 1)(y + 1) Remove parenthesis around (5 + -5y) 5 + -5y + -20 = -1(5y + 1)(y + 1) Reorder the terms: 5 + -20 + -5y = -1(5y + 1)(y + 1) Combine like terms: 5 + -20 = -15 -15 + -5y = -1(5y + 1)(y + 1) Reorder the terms: -15 + -5y = -1(1 + 5y)(y + 1) Reorder the terms: -15 + -5y = -1(1 + 5y)(1 + y) Multiply (1 + 5y) * (1 + y) -15 + -5y = -1(1(1 + y) + 5y * (1 + y)) -15 + -5y = -1((1 * 1 + y * 1) + 5y * (1 + y)) -15 + -5y = -1((1 + 1y) + 5y * (1 + y)) -15 + -5y = -1(1 + 1y + (1 * 5y + y * 5y)) -15 + -5y = -1(1 + 1y + (5y + 5y2)) Combine like terms: 1y + 5y = 6y -15 + -5y = -1(1 + 6y + 5y2) -15 + -5y = (1 * -1 + 6y * -1 + 5y2 * -1) -15 + -5y = (-1 + -6y + -5y2) Solving -15 + -5y = -1 + -6y + -5y2 Solving for variable 'y'. Reorder the terms: -15 + 1 + -5y + 6y + 5y2 = -1 + -6y + -5y2 + 1 + 6y + 5y2 Combine like terms: -15 + 1 = -14 -14 + -5y + 6y + 5y2 = -1 + -6y + -5y2 + 1 + 6y + 5y2 Combine like terms: -5y + 6y = 1y -14 + 1y + 5y2 = -1 + -6y + -5y2 + 1 + 6y + 5y2 Reorder the terms: -14 + 1y + 5y2 = -1 + 1 + -6y + 6y + -5y2 + 5y2 Combine like terms: -1 + 1 = 0 -14 + 1y + 5y2 = 0 + -6y + 6y + -5y2 + 5y2 -14 + 1y + 5y2 = -6y + 6y + -5y2 + 5y2 Combine like terms: -6y + 6y = 0 -14 + 1y + 5y2 = 0 + -5y2 + 5y2 -14 + 1y + 5y2 = -5y2 + 5y2 Combine like terms: -5y2 + 5y2 = 0 -14 + 1y + 5y2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -2.8 + 0.2y + y2 = 0 Move the constant term to the right: Add '2.8' to each side of the equation. -2.8 + 0.2y + 2.8 + y2 = 0 + 2.8 Reorder the terms: -2.8 + 2.8 + 0.2y + y2 = 0 + 2.8 Combine like terms: -2.8 + 2.8 = 0.0 0.0 + 0.2y + y2 = 0 + 2.8 0.2y + y2 = 0 + 2.8 Combine like terms: 0 + 2.8 = 2.8 0.2y + y2 = 2.8 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 = 2.8 + 0.01 Reorder the terms: 0.01 + 0.2y + y2 = 2.8 + 0.01 Combine like terms: 2.8 + 0.01 = 2.81 0.01 + 0.2y + y2 = 2.81 Factor a perfect square on the left side: (y + 0.1)(y + 0.1) = 2.81 Calculate the square root of the right side: 1.676305461 Break this problem into two subproblems by setting (y + 0.1) equal to 1.676305461 and -1.676305461.Subproblem 1
y + 0.1 = 1.676305461 Simplifying y + 0.1 = 1.676305461 Reorder the terms: 0.1 + y = 1.676305461 Solving 0.1 + y = 1.676305461 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 = 1.676305461 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + y = 1.676305461 + -0.1 y = 1.676305461 + -0.1 Combine like terms: 1.676305461 + -0.1 = 1.576305461 y = 1.576305461 Simplifying y = 1.576305461Subproblem 2
y + 0.1 = -1.676305461 Simplifying y + 0.1 = -1.676305461 Reorder the terms: 0.1 + y = -1.676305461 Solving 0.1 + y = -1.676305461 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 = -1.676305461 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + y = -1.676305461 + -0.1 y = -1.676305461 + -0.1 Combine like terms: -1.676305461 + -0.1 = -1.776305461 y = -1.776305461 Simplifying y = -1.776305461Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.576305461, -1.776305461}
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