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Simplifying 2y(8 + y) = 77 (8 * 2y + y * 2y) = 77 (16y + 2y2) = 77 Solving 16y + 2y2 = 77 Solving for variable 'y'. Reorder the terms: -77 + 16y + 2y2 = 77 + -77 Combine like terms: 77 + -77 = 0 -77 + 16y + 2y2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -38.5 + 8y + y2 = 0 Move the constant term to the right: Add '38.5' to each side of the equation. -38.5 + 8y + 38.5 + y2 = 0 + 38.5 Reorder the terms: -38.5 + 38.5 + 8y + y2 = 0 + 38.5 Combine like terms: -38.5 + 38.5 = 0.0 0.0 + 8y + y2 = 0 + 38.5 8y + y2 = 0 + 38.5 Combine like terms: 0 + 38.5 = 38.5 8y + y2 = 38.5 The y term is 8y. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8y + 16 + y2 = 38.5 + 16 Reorder the terms: 16 + 8y + y2 = 38.5 + 16 Combine like terms: 38.5 + 16 = 54.5 16 + 8y + y2 = 54.5 Factor a perfect square on the left side: (y + 4)(y + 4) = 54.5 Calculate the square root of the right side: 7.38241153 Break this problem into two subproblems by setting (y + 4) equal to 7.38241153 and -7.38241153.Subproblem 1
y + 4 = 7.38241153 Simplifying y + 4 = 7.38241153 Reorder the terms: 4 + y = 7.38241153 Solving 4 + y = 7.38241153 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + y = 7.38241153 + -4 Combine like terms: 4 + -4 = 0 0 + y = 7.38241153 + -4 y = 7.38241153 + -4 Combine like terms: 7.38241153 + -4 = 3.38241153 y = 3.38241153 Simplifying y = 3.38241153Subproblem 2
y + 4 = -7.38241153 Simplifying y + 4 = -7.38241153 Reorder the terms: 4 + y = -7.38241153 Solving 4 + y = -7.38241153 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + y = -7.38241153 + -4 Combine like terms: 4 + -4 = 0 0 + y = -7.38241153 + -4 y = -7.38241153 + -4 Combine like terms: -7.38241153 + -4 = -11.38241153 y = -11.38241153 Simplifying y = -11.38241153Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.38241153, -11.38241153}
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