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Simplifying 1000 + -45t + 0.2t2 = 0 Solving 1000 + -45t + 0.2t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 0.2 the coefficient of the squared term: Divide each side by '0.2'. 5000 + -225t + t2 = 0 Move the constant term to the right: Add '-5000' to each side of the equation. 5000 + -225t + -5000 + t2 = 0 + -5000 Reorder the terms: 5000 + -5000 + -225t + t2 = 0 + -5000 Combine like terms: 5000 + -5000 = 0 0 + -225t + t2 = 0 + -5000 -225t + t2 = 0 + -5000 Combine like terms: 0 + -5000 = -5000 -225t + t2 = -5000 The t term is -225t. Take half its coefficient (-112.5). Square it (12656.25) and add it to both sides. Add '12656.25' to each side of the equation. -225t + 12656.25 + t2 = -5000 + 12656.25 Reorder the terms: 12656.25 + -225t + t2 = -5000 + 12656.25 Combine like terms: -5000 + 12656.25 = 7656.25 12656.25 + -225t + t2 = 7656.25 Factor a perfect square on the left side: (t + -112.5)(t + -112.5) = 7656.25 Calculate the square root of the right side: 87.5 Break this problem into two subproblems by setting (t + -112.5) equal to 87.5 and -87.5.Subproblem 1
t + -112.5 = 87.5 Simplifying t + -112.5 = 87.5 Reorder the terms: -112.5 + t = 87.5 Solving -112.5 + t = 87.5 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '112.5' to each side of the equation. -112.5 + 112.5 + t = 87.5 + 112.5 Combine like terms: -112.5 + 112.5 = 0.0 0.0 + t = 87.5 + 112.5 t = 87.5 + 112.5 Combine like terms: 87.5 + 112.5 = 200 t = 200 Simplifying t = 200Subproblem 2
t + -112.5 = -87.5 Simplifying t + -112.5 = -87.5 Reorder the terms: -112.5 + t = -87.5 Solving -112.5 + t = -87.5 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '112.5' to each side of the equation. -112.5 + 112.5 + t = -87.5 + 112.5 Combine like terms: -112.5 + 112.5 = 0.0 0.0 + t = -87.5 + 112.5 t = -87.5 + 112.5 Combine like terms: -87.5 + 112.5 = 25 t = 25 Simplifying t = 25Solution
The solution to the problem is based on the solutions from the subproblems. t = {200, 25}
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