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Simplifying y(2y + 4) = 77 Reorder the terms: y(4 + 2y) = 77 (4 * y + 2y * y) = 77 (4y + 2y2) = 77 Solving 4y + 2y2 = 77 Solving for variable 'y'. Reorder the terms: -77 + 4y + 2y2 = 77 + -77 Combine like terms: 77 + -77 = 0 -77 + 4y + 2y2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -38.5 + 2y + y2 = 0 Move the constant term to the right: Add '38.5' to each side of the equation. -38.5 + 2y + 38.5 + y2 = 0 + 38.5 Reorder the terms: -38.5 + 38.5 + 2y + y2 = 0 + 38.5 Combine like terms: -38.5 + 38.5 = 0.0 0.0 + 2y + y2 = 0 + 38.5 2y + y2 = 0 + 38.5 Combine like terms: 0 + 38.5 = 38.5 2y + y2 = 38.5 The y term is 2y. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2y + 1 + y2 = 38.5 + 1 Reorder the terms: 1 + 2y + y2 = 38.5 + 1 Combine like terms: 38.5 + 1 = 39.5 1 + 2y + y2 = 39.5 Factor a perfect square on the left side: (y + 1)(y + 1) = 39.5 Calculate the square root of the right side: 6.284902545 Break this problem into two subproblems by setting (y + 1) equal to 6.284902545 and -6.284902545.Subproblem 1
y + 1 = 6.284902545 Simplifying y + 1 = 6.284902545 Reorder the terms: 1 + y = 6.284902545 Solving 1 + y = 6.284902545 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = 6.284902545 + -1 Combine like terms: 1 + -1 = 0 0 + y = 6.284902545 + -1 y = 6.284902545 + -1 Combine like terms: 6.284902545 + -1 = 5.284902545 y = 5.284902545 Simplifying y = 5.284902545Subproblem 2
y + 1 = -6.284902545 Simplifying y + 1 = -6.284902545 Reorder the terms: 1 + y = -6.284902545 Solving 1 + y = -6.284902545 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = -6.284902545 + -1 Combine like terms: 1 + -1 = 0 0 + y = -6.284902545 + -1 y = -6.284902545 + -1 Combine like terms: -6.284902545 + -1 = -7.284902545 y = -7.284902545 Simplifying y = -7.284902545Solution
The solution to the problem is based on the solutions from the subproblems. y = {5.284902545, -7.284902545}
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