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Simplifying 142104 = 563t + -0.5t2 Solving 142104 = 563t + -0.5t2 Solving for variable 't'. Reorder the terms: 142104 + -563t + 0.5t2 = 563t + -563t + -0.5t2 + 0.5t2 Combine like terms: 563t + -563t = 0 142104 + -563t + 0.5t2 = 0 + -0.5t2 + 0.5t2 142104 + -563t + 0.5t2 = -0.5t2 + 0.5t2 Combine like terms: -0.5t2 + 0.5t2 = 0.0 142104 + -563t + 0.5t2 = 0.0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. 284208 + -1126t + t2 = 0 Move the constant term to the right: Add '-284208' to each side of the equation. 284208 + -1126t + -284208 + t2 = 0 + -284208 Reorder the terms: 284208 + -284208 + -1126t + t2 = 0 + -284208 Combine like terms: 284208 + -284208 = 0 0 + -1126t + t2 = 0 + -284208 -1126t + t2 = 0 + -284208 Combine like terms: 0 + -284208 = -284208 -1126t + t2 = -284208 The t term is -1126t. Take half its coefficient (-563). Square it (316969) and add it to both sides. Add '316969' to each side of the equation. -1126t + 316969 + t2 = -284208 + 316969 Reorder the terms: 316969 + -1126t + t2 = -284208 + 316969 Combine like terms: -284208 + 316969 = 32761 316969 + -1126t + t2 = 32761 Factor a perfect square on the left side: (t + -563)(t + -563) = 32761 Calculate the square root of the right side: 181 Break this problem into two subproblems by setting (t + -563) equal to 181 and -181.Subproblem 1
t + -563 = 181 Simplifying t + -563 = 181 Reorder the terms: -563 + t = 181 Solving -563 + t = 181 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '563' to each side of the equation. -563 + 563 + t = 181 + 563 Combine like terms: -563 + 563 = 0 0 + t = 181 + 563 t = 181 + 563 Combine like terms: 181 + 563 = 744 t = 744 Simplifying t = 744Subproblem 2
t + -563 = -181 Simplifying t + -563 = -181 Reorder the terms: -563 + t = -181 Solving -563 + t = -181 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '563' to each side of the equation. -563 + 563 + t = -181 + 563 Combine like terms: -563 + 563 = 0 0 + t = -181 + 563 t = -181 + 563 Combine like terms: -181 + 563 = 382 t = 382 Simplifying t = 382Solution
The solution to the problem is based on the solutions from the subproblems. t = {744, 382}
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