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Simplifying 15 + 13t(84 + -1t) = 84 15 + (84 * 13t + -1t * 13t) = 84 15 + (1092t + -13t2) = 84 Solving 15 + 1092t + -13t2 = 84 Solving for variable 't'. Reorder the terms: 15 + -84 + 1092t + -13t2 = 84 + -84 Combine like terms: 15 + -84 = -69 -69 + 1092t + -13t2 = 84 + -84 Combine like terms: 84 + -84 = 0 -69 + 1092t + -13t2 = 0 Begin completing the square. Divide all terms by -13 the coefficient of the squared term: Divide each side by '-13'. 5.307692308 + -84t + t2 = 0 Move the constant term to the right: Add '-5.307692308' to each side of the equation. 5.307692308 + -84t + -5.307692308 + t2 = 0 + -5.307692308 Reorder the terms: 5.307692308 + -5.307692308 + -84t + t2 = 0 + -5.307692308 Combine like terms: 5.307692308 + -5.307692308 = 0.000000000 0.000000000 + -84t + t2 = 0 + -5.307692308 -84t + t2 = 0 + -5.307692308 Combine like terms: 0 + -5.307692308 = -5.307692308 -84t + t2 = -5.307692308 The t term is -84t. Take half its coefficient (-42). Square it (1764) and add it to both sides. Add '1764' to each side of the equation. -84t + 1764 + t2 = -5.307692308 + 1764 Reorder the terms: 1764 + -84t + t2 = -5.307692308 + 1764 Combine like terms: -5.307692308 + 1764 = 1758.692307692 1764 + -84t + t2 = 1758.692307692 Factor a perfect square on the left side: (t + -42)(t + -42) = 1758.692307692 Calculate the square root of the right side: 41.936765585 Break this problem into two subproblems by setting (t + -42) equal to 41.936765585 and -41.936765585.Subproblem 1
t + -42 = 41.936765585 Simplifying t + -42 = 41.936765585 Reorder the terms: -42 + t = 41.936765585 Solving -42 + t = 41.936765585 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '42' to each side of the equation. -42 + 42 + t = 41.936765585 + 42 Combine like terms: -42 + 42 = 0 0 + t = 41.936765585 + 42 t = 41.936765585 + 42 Combine like terms: 41.936765585 + 42 = 83.936765585 t = 83.936765585 Simplifying t = 83.936765585Subproblem 2
t + -42 = -41.936765585 Simplifying t + -42 = -41.936765585 Reorder the terms: -42 + t = -41.936765585 Solving -42 + t = -41.936765585 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '42' to each side of the equation. -42 + 42 + t = -41.936765585 + 42 Combine like terms: -42 + 42 = 0 0 + t = -41.936765585 + 42 t = -41.936765585 + 42 Combine like terms: -41.936765585 + 42 = 0.063234415 t = 0.063234415 Simplifying t = 0.063234415Solution
The solution to the problem is based on the solutions from the subproblems. t = {83.936765585, 0.063234415}
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