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Simplifying (h + -3)(h + -5) = 1 Reorder the terms: (-3 + h)(h + -5) = 1 Reorder the terms: (-3 + h)(-5 + h) = 1 Multiply (-3 + h) * (-5 + h) (-3(-5 + h) + h(-5 + h)) = 1 ((-5 * -3 + h * -3) + h(-5 + h)) = 1 ((15 + -3h) + h(-5 + h)) = 1 (15 + -3h + (-5 * h + h * h)) = 1 (15 + -3h + (-5h + h2)) = 1 Combine like terms: -3h + -5h = -8h (15 + -8h + h2) = 1 Solving 15 + -8h + h2 = 1 Solving for variable 'h'. Reorder the terms: 15 + -1 + -8h + h2 = 1 + -1 Combine like terms: 15 + -1 = 14 14 + -8h + h2 = 1 + -1 Combine like terms: 1 + -1 = 0 14 + -8h + h2 = 0 Begin completing the square. Move the constant term to the right: Add '-14' to each side of the equation. 14 + -8h + -14 + h2 = 0 + -14 Reorder the terms: 14 + -14 + -8h + h2 = 0 + -14 Combine like terms: 14 + -14 = 0 0 + -8h + h2 = 0 + -14 -8h + h2 = 0 + -14 Combine like terms: 0 + -14 = -14 -8h + h2 = -14 The h term is -8h. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8h + 16 + h2 = -14 + 16 Reorder the terms: 16 + -8h + h2 = -14 + 16 Combine like terms: -14 + 16 = 2 16 + -8h + h2 = 2 Factor a perfect square on the left side: (h + -4)(h + -4) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (h + -4) equal to 1.414213562 and -1.414213562.Subproblem 1
h + -4 = 1.414213562 Simplifying h + -4 = 1.414213562 Reorder the terms: -4 + h = 1.414213562 Solving -4 + h = 1.414213562 Solving for variable 'h'. Move all terms containing h to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + h = 1.414213562 + 4 Combine like terms: -4 + 4 = 0 0 + h = 1.414213562 + 4 h = 1.414213562 + 4 Combine like terms: 1.414213562 + 4 = 5.414213562 h = 5.414213562 Simplifying h = 5.414213562Subproblem 2
h + -4 = -1.414213562 Simplifying h + -4 = -1.414213562 Reorder the terms: -4 + h = -1.414213562 Solving -4 + h = -1.414213562 Solving for variable 'h'. Move all terms containing h to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + h = -1.414213562 + 4 Combine like terms: -4 + 4 = 0 0 + h = -1.414213562 + 4 h = -1.414213562 + 4 Combine like terms: -1.414213562 + 4 = 2.585786438 h = 2.585786438 Simplifying h = 2.585786438Solution
The solution to the problem is based on the solutions from the subproblems. h = {5.414213562, 2.585786438}
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