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Simplifying 77 = h(h + 3) Reorder the terms: 77 = h(3 + h) 77 = (3 * h + h * h) 77 = (3h + h2) Solving 77 = 3h + h2 Solving for variable 'h'. Reorder the terms: 77 + -3h + -1h2 = 3h + -3h + h2 + -1h2 Combine like terms: 3h + -3h = 0 77 + -3h + -1h2 = 0 + h2 + -1h2 77 + -3h + -1h2 = h2 + -1h2 Combine like terms: h2 + -1h2 = 0 77 + -3h + -1h2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -77 + 3h + h2 = 0 Move the constant term to the right: Add '77' to each side of the equation. -77 + 3h + 77 + h2 = 0 + 77 Reorder the terms: -77 + 77 + 3h + h2 = 0 + 77 Combine like terms: -77 + 77 = 0 0 + 3h + h2 = 0 + 77 3h + h2 = 0 + 77 Combine like terms: 0 + 77 = 77 3h + h2 = 77 The h term is 3h. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3h + 2.25 + h2 = 77 + 2.25 Reorder the terms: 2.25 + 3h + h2 = 77 + 2.25 Combine like terms: 77 + 2.25 = 79.25 2.25 + 3h + h2 = 79.25 Factor a perfect square on the left side: (h + 1.5)(h + 1.5) = 79.25 Calculate the square root of the right side: 8.902246907 Break this problem into two subproblems by setting (h + 1.5) equal to 8.902246907 and -8.902246907.Subproblem 1
h + 1.5 = 8.902246907 Simplifying h + 1.5 = 8.902246907 Reorder the terms: 1.5 + h = 8.902246907 Solving 1.5 + h = 8.902246907 Solving for variable 'h'. Move all terms containing h to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + h = 8.902246907 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + h = 8.902246907 + -1.5 h = 8.902246907 + -1.5 Combine like terms: 8.902246907 + -1.5 = 7.402246907 h = 7.402246907 Simplifying h = 7.402246907Subproblem 2
h + 1.5 = -8.902246907 Simplifying h + 1.5 = -8.902246907 Reorder the terms: 1.5 + h = -8.902246907 Solving 1.5 + h = -8.902246907 Solving for variable 'h'. Move all terms containing h to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + h = -8.902246907 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + h = -8.902246907 + -1.5 h = -8.902246907 + -1.5 Combine like terms: -8.902246907 + -1.5 = -10.402246907 h = -10.402246907 Simplifying h = -10.402246907Solution
The solution to the problem is based on the solutions from the subproblems. h = {7.402246907, -10.402246907}
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