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Simplifying 165 = (3h + 2) * h Reorder the terms: 165 = (2 + 3h) * h Reorder the terms for easier multiplication: 165 = h(2 + 3h) 165 = (2 * h + 3h * h) 165 = (2h + 3h2) Solving 165 = 2h + 3h2 Solving for variable 'h'. Reorder the terms: 165 + -2h + -3h2 = 2h + -2h + 3h2 + -3h2 Combine like terms: 2h + -2h = 0 165 + -2h + -3h2 = 0 + 3h2 + -3h2 165 + -2h + -3h2 = 3h2 + -3h2 Combine like terms: 3h2 + -3h2 = 0 165 + -2h + -3h2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -55 + 0.6666666667h + h2 = 0 Move the constant term to the right: Add '55' to each side of the equation. -55 + 0.6666666667h + 55 + h2 = 0 + 55 Reorder the terms: -55 + 55 + 0.6666666667h + h2 = 0 + 55 Combine like terms: -55 + 55 = 0 0 + 0.6666666667h + h2 = 0 + 55 0.6666666667h + h2 = 0 + 55 Combine like terms: 0 + 55 = 55 0.6666666667h + h2 = 55 The h term is 0.6666666667h. Take half its coefficient (0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. 0.6666666667h + 0.1111111112 + h2 = 55 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667h + h2 = 55 + 0.1111111112 Combine like terms: 55 + 0.1111111112 = 55.1111111112 0.1111111112 + 0.6666666667h + h2 = 55.1111111112 Factor a perfect square on the left side: (h + 0.3333333334)(h + 0.3333333334) = 55.1111111112 Calculate the square root of the right side: 7.423685817 Break this problem into two subproblems by setting (h + 0.3333333334) equal to 7.423685817 and -7.423685817.Subproblem 1
h + 0.3333333334 = 7.423685817 Simplifying h + 0.3333333334 = 7.423685817 Reorder the terms: 0.3333333334 + h = 7.423685817 Solving 0.3333333334 + h = 7.423685817 Solving for variable 'h'. Move all terms containing h to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + h = 7.423685817 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + h = 7.423685817 + -0.3333333334 h = 7.423685817 + -0.3333333334 Combine like terms: 7.423685817 + -0.3333333334 = 7.0903524836 h = 7.0903524836 Simplifying h = 7.0903524836Subproblem 2
h + 0.3333333334 = -7.423685817 Simplifying h + 0.3333333334 = -7.423685817 Reorder the terms: 0.3333333334 + h = -7.423685817 Solving 0.3333333334 + h = -7.423685817 Solving for variable 'h'. Move all terms containing h to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + h = -7.423685817 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + h = -7.423685817 + -0.3333333334 h = -7.423685817 + -0.3333333334 Combine like terms: -7.423685817 + -0.3333333334 = -7.7570191504 h = -7.7570191504 Simplifying h = -7.7570191504Solution
The solution to the problem is based on the solutions from the subproblems. h = {7.0903524836, -7.7570191504}
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