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Simplifying 4(10 + -2.5y) * 10y = 40 Reorder the terms for easier multiplication: 4 * 10y(10 + -2.5y) = 40 Multiply 4 * 10 40y(10 + -2.5y) = 40 (10 * 40y + -2.5y * 40y) = 40 (400y + -100y2) = 40 Solving 400y + -100y2 = 40 Solving for variable 'y'. Reorder the terms: -40 + 400y + -100y2 = 40 + -40 Combine like terms: 40 + -40 = 0 -40 + 400y + -100y2 = 0 Factor out the Greatest Common Factor (GCF), '20'. 20(-2 + 20y + -5y2) = 0 Ignore the factor 20.Subproblem 1
Set the factor '(-2 + 20y + -5y2)' equal to zero and attempt to solve: Simplifying -2 + 20y + -5y2 = 0 Solving -2 + 20y + -5y2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. 0.4 + -4y + y2 = 0 Move the constant term to the right: Add '-0.4' to each side of the equation. 0.4 + -4y + -0.4 + y2 = 0 + -0.4 Reorder the terms: 0.4 + -0.4 + -4y + y2 = 0 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + -4y + y2 = 0 + -0.4 -4y + y2 = 0 + -0.4 Combine like terms: 0 + -0.4 = -0.4 -4y + y2 = -0.4 The y term is -4y. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4y + 4 + y2 = -0.4 + 4 Reorder the terms: 4 + -4y + y2 = -0.4 + 4 Combine like terms: -0.4 + 4 = 3.6 4 + -4y + y2 = 3.6 Factor a perfect square on the left side: (y + -2)(y + -2) = 3.6 Calculate the square root of the right side: 1.897366596 Break this problem into two subproblems by setting (y + -2) equal to 1.897366596 and -1.897366596.Subproblem 1
y + -2 = 1.897366596 Simplifying y + -2 = 1.897366596 Reorder the terms: -2 + y = 1.897366596 Solving -2 + y = 1.897366596 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + y = 1.897366596 + 2 Combine like terms: -2 + 2 = 0 0 + y = 1.897366596 + 2 y = 1.897366596 + 2 Combine like terms: 1.897366596 + 2 = 3.897366596 y = 3.897366596 Simplifying y = 3.897366596Subproblem 2
y + -2 = -1.897366596 Simplifying y + -2 = -1.897366596 Reorder the terms: -2 + y = -1.897366596 Solving -2 + y = -1.897366596 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + y = -1.897366596 + 2 Combine like terms: -2 + 2 = 0 0 + y = -1.897366596 + 2 y = -1.897366596 + 2 Combine like terms: -1.897366596 + 2 = 0.102633404 y = 0.102633404 Simplifying y = 0.102633404Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.897366596, 0.102633404}Solution
y = {3.897366596, 0.102633404}
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