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Simplifying 0.5[(10 + h)(h)] = 12 Reorder the terms for easier multiplication: 0.5[h(10 + h)] = 12 0.5[(10 * h + h * h)] = 12 0.5[(10h + h2)] = 12 [10h * 0.5 + h2 * 0.5] = 12 [5h + 0.5h2] = 12 Solving 5h + 0.5h2 = 12 Solving for variable 'h'. Reorder the terms: -12 + 5h + 0.5h2 = 12 + -12 Combine like terms: 12 + -12 = 0 -12 + 5h + 0.5h2 = 0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -24 + 10h + h2 = 0 Move the constant term to the right: Add '24' to each side of the equation. -24 + 10h + 24 + h2 = 0 + 24 Reorder the terms: -24 + 24 + 10h + h2 = 0 + 24 Combine like terms: -24 + 24 = 0 0 + 10h + h2 = 0 + 24 10h + h2 = 0 + 24 Combine like terms: 0 + 24 = 24 10h + h2 = 24 The h term is 10h. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10h + 25 + h2 = 24 + 25 Reorder the terms: 25 + 10h + h2 = 24 + 25 Combine like terms: 24 + 25 = 49 25 + 10h + h2 = 49 Factor a perfect square on the left side: (h + 5)(h + 5) = 49 Calculate the square root of the right side: 7 Break this problem into two subproblems by setting (h + 5) equal to 7 and -7.Subproblem 1
h + 5 = 7 Simplifying h + 5 = 7 Reorder the terms: 5 + h = 7 Solving 5 + h = 7 Solving for variable 'h'. Move all terms containing h to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + h = 7 + -5 Combine like terms: 5 + -5 = 0 0 + h = 7 + -5 h = 7 + -5 Combine like terms: 7 + -5 = 2 h = 2 Simplifying h = 2Subproblem 2
h + 5 = -7 Simplifying h + 5 = -7 Reorder the terms: 5 + h = -7 Solving 5 + h = -7 Solving for variable 'h'. Move all terms containing h to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + h = -7 + -5 Combine like terms: 5 + -5 = 0 0 + h = -7 + -5 h = -7 + -5 Combine like terms: -7 + -5 = -12 h = -12 Simplifying h = -12Solution
The solution to the problem is based on the solutions from the subproblems. h = {2, -12}
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