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Simplifying t2 + 8t + -8084 = 2446096 Reorder the terms: -8084 + 8t + t2 = 2446096 Solving -8084 + 8t + t2 = 2446096 Solving for variable 't'. Reorder the terms: -8084 + -2446096 + 8t + t2 = 2446096 + -2446096 Combine like terms: -8084 + -2446096 = -2454180 -2454180 + 8t + t2 = 2446096 + -2446096 Combine like terms: 2446096 + -2446096 = 0 -2454180 + 8t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '2454180' to each side of the equation. -2454180 + 8t + 2454180 + t2 = 0 + 2454180 Reorder the terms: -2454180 + 2454180 + 8t + t2 = 0 + 2454180 Combine like terms: -2454180 + 2454180 = 0 0 + 8t + t2 = 0 + 2454180 8t + t2 = 0 + 2454180 Combine like terms: 0 + 2454180 = 2454180 8t + t2 = 2454180 The t term is 8t. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8t + 16 + t2 = 2454180 + 16 Reorder the terms: 16 + 8t + t2 = 2454180 + 16 Combine like terms: 2454180 + 16 = 2454196 16 + 8t + t2 = 2454196 Factor a perfect square on the left side: (t + 4)(t + 4) = 2454196 Calculate the square root of the right side: 1566.58737388 Break this problem into two subproblems by setting (t + 4) equal to 1566.58737388 and -1566.58737388.Subproblem 1
t + 4 = 1566.58737388 Simplifying t + 4 = 1566.58737388 Reorder the terms: 4 + t = 1566.58737388 Solving 4 + t = 1566.58737388 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + t = 1566.58737388 + -4 Combine like terms: 4 + -4 = 0 0 + t = 1566.58737388 + -4 t = 1566.58737388 + -4 Combine like terms: 1566.58737388 + -4 = 1562.58737388 t = 1562.58737388 Simplifying t = 1562.58737388Subproblem 2
t + 4 = -1566.58737388 Simplifying t + 4 = -1566.58737388 Reorder the terms: 4 + t = -1566.58737388 Solving 4 + t = -1566.58737388 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + t = -1566.58737388 + -4 Combine like terms: 4 + -4 = 0 0 + t = -1566.58737388 + -4 t = -1566.58737388 + -4 Combine like terms: -1566.58737388 + -4 = -1570.58737388 t = -1570.58737388 Simplifying t = -1570.58737388Solution
The solution to the problem is based on the solutions from the subproblems. t = {1562.58737388, -1570.58737388}
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