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Simplifying 24 = 0.5(h2 + 2h) Reorder the terms: 24 = 0.5(2h + h2) 24 = (2h * 0.5 + h2 * 0.5) 24 = (1h + 0.5h2) Solving 24 = 1h + 0.5h2 Solving for variable 'h'. Reorder the terms: 24 + -1h + -0.5h2 = 1h + -1h + 0.5h2 + -0.5h2 Combine like terms: 1h + -1h = 0 24 + -1h + -0.5h2 = 0 + 0.5h2 + -0.5h2 24 + -1h + -0.5h2 = 0.5h2 + -0.5h2 Combine like terms: 0.5h2 + -0.5h2 = 0.0 24 + -1h + -0.5h2 = 0.0 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. -48 + 2h + h2 = 0 Move the constant term to the right: Add '48' to each side of the equation. -48 + 2h + 48 + h2 = 0 + 48 Reorder the terms: -48 + 48 + 2h + h2 = 0 + 48 Combine like terms: -48 + 48 = 0 0 + 2h + h2 = 0 + 48 2h + h2 = 0 + 48 Combine like terms: 0 + 48 = 48 2h + h2 = 48 The h term is 2h. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2h + 1 + h2 = 48 + 1 Reorder the terms: 1 + 2h + h2 = 48 + 1 Combine like terms: 48 + 1 = 49 1 + 2h + h2 = 49 Factor a perfect square on the left side: (h + 1)(h + 1) = 49 Calculate the square root of the right side: 7 Break this problem into two subproblems by setting (h + 1) equal to 7 and -7.Subproblem 1
h + 1 = 7 Simplifying h + 1 = 7 Reorder the terms: 1 + h = 7 Solving 1 + h = 7 Solving for variable 'h'. Move all terms containing h to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + h = 7 + -1 Combine like terms: 1 + -1 = 0 0 + h = 7 + -1 h = 7 + -1 Combine like terms: 7 + -1 = 6 h = 6 Simplifying h = 6Subproblem 2
h + 1 = -7 Simplifying h + 1 = -7 Reorder the terms: 1 + h = -7 Solving 1 + h = -7 Solving for variable 'h'. Move all terms containing h to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + h = -7 + -1 Combine like terms: 1 + -1 = 0 0 + h = -7 + -1 h = -7 + -1 Combine like terms: -7 + -1 = -8 h = -8 Simplifying h = -8Solution
The solution to the problem is based on the solutions from the subproblems. h = {6, -8}
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