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Simplifying h(3h + 1) = 180 Reorder the terms: h(1 + 3h) = 180 (1 * h + 3h * h) = 180 (1h + 3h2) = 180 Solving 1h + 3h2 = 180 Solving for variable 'h'. Reorder the terms: -180 + 1h + 3h2 = 180 + -180 Combine like terms: 180 + -180 = 0 -180 + 1h + 3h2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -60 + 0.3333333333h + h2 = 0 Move the constant term to the right: Add '60' to each side of the equation. -60 + 0.3333333333h + 60 + h2 = 0 + 60 Reorder the terms: -60 + 60 + 0.3333333333h + h2 = 0 + 60 Combine like terms: -60 + 60 = 0 0 + 0.3333333333h + h2 = 0 + 60 0.3333333333h + h2 = 0 + 60 Combine like terms: 0 + 60 = 60 0.3333333333h + h2 = 60 The h term is 0.3333333333h. Take half its coefficient (0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. 0.3333333333h + 0.02777777779 + h2 = 60 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333h + h2 = 60 + 0.02777777779 Combine like terms: 60 + 0.02777777779 = 60.02777777779 0.02777777779 + 0.3333333333h + h2 = 60.02777777779 Factor a perfect square on the left side: (h + 0.1666666667)(h + 0.1666666667) = 60.02777777779 Calculate the square root of the right side: 7.747759533 Break this problem into two subproblems by setting (h + 0.1666666667) equal to 7.747759533 and -7.747759533.Subproblem 1
h + 0.1666666667 = 7.747759533 Simplifying h + 0.1666666667 = 7.747759533 Reorder the terms: 0.1666666667 + h = 7.747759533 Solving 0.1666666667 + h = 7.747759533 Solving for variable 'h'. Move all terms containing h to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + h = 7.747759533 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + h = 7.747759533 + -0.1666666667 h = 7.747759533 + -0.1666666667 Combine like terms: 7.747759533 + -0.1666666667 = 7.5810928663 h = 7.5810928663 Simplifying h = 7.5810928663Subproblem 2
h + 0.1666666667 = -7.747759533 Simplifying h + 0.1666666667 = -7.747759533 Reorder the terms: 0.1666666667 + h = -7.747759533 Solving 0.1666666667 + h = -7.747759533 Solving for variable 'h'. Move all terms containing h to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + h = -7.747759533 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + h = -7.747759533 + -0.1666666667 h = -7.747759533 + -0.1666666667 Combine like terms: -7.747759533 + -0.1666666667 = -7.9144261997 h = -7.9144261997 Simplifying h = -7.9144261997Solution
The solution to the problem is based on the solutions from the subproblems. h = {7.5810928663, -7.9144261997}
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