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Simplifying 7(4y + -1) * 2y = 1 Reorder the terms: 7(-1 + 4y) * 2y = 1 Reorder the terms for easier multiplication: 7 * 2y(-1 + 4y) = 1 Multiply 7 * 2 14y(-1 + 4y) = 1 (-1 * 14y + 4y * 14y) = 1 (-14y + 56y2) = 1 Solving -14y + 56y2 = 1 Solving for variable 'y'. Reorder the terms: -1 + -14y + 56y2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + -14y + 56y2 = 0 Begin completing the square. Divide all terms by 56 the coefficient of the squared term: Divide each side by '56'. -0.01785714286 + -0.25y + y2 = 0 Move the constant term to the right: Add '0.01785714286' to each side of the equation. -0.01785714286 + -0.25y + 0.01785714286 + y2 = 0 + 0.01785714286 Reorder the terms: -0.01785714286 + 0.01785714286 + -0.25y + y2 = 0 + 0.01785714286 Combine like terms: -0.01785714286 + 0.01785714286 = 0.00000000000 0.00000000000 + -0.25y + y2 = 0 + 0.01785714286 -0.25y + y2 = 0 + 0.01785714286 Combine like terms: 0 + 0.01785714286 = 0.01785714286 -0.25y + y2 = 0.01785714286 The y term is -0.25y. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25y + 0.015625 + y2 = 0.01785714286 + 0.015625 Reorder the terms: 0.015625 + -0.25y + y2 = 0.01785714286 + 0.015625 Combine like terms: 0.01785714286 + 0.015625 = 0.03348214286 0.015625 + -0.25y + y2 = 0.03348214286 Factor a perfect square on the left side: (y + -0.125)(y + -0.125) = 0.03348214286 Calculate the square root of the right side: 0.182981264 Break this problem into two subproblems by setting (y + -0.125) equal to 0.182981264 and -0.182981264.Subproblem 1
y + -0.125 = 0.182981264 Simplifying y + -0.125 = 0.182981264 Reorder the terms: -0.125 + y = 0.182981264 Solving -0.125 + y = 0.182981264 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + y = 0.182981264 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + y = 0.182981264 + 0.125 y = 0.182981264 + 0.125 Combine like terms: 0.182981264 + 0.125 = 0.307981264 y = 0.307981264 Simplifying y = 0.307981264Subproblem 2
y + -0.125 = -0.182981264 Simplifying y + -0.125 = -0.182981264 Reorder the terms: -0.125 + y = -0.182981264 Solving -0.125 + y = -0.182981264 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + y = -0.182981264 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + y = -0.182981264 + 0.125 y = -0.182981264 + 0.125 Combine like terms: -0.182981264 + 0.125 = -0.057981264 y = -0.057981264 Simplifying y = -0.057981264Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.307981264, -0.057981264}
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