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Simplifying 45 = 16t2 + 55t + 6 Reorder the terms: 45 = 6 + 55t + 16t2 Solving 45 = 6 + 55t + 16t2 Solving for variable 't'. Combine like terms: 45 + -6 = 39 39 + -55t + -16t2 = 6 + 55t + 16t2 + -6 + -55t + -16t2 Reorder the terms: 39 + -55t + -16t2 = 6 + -6 + 55t + -55t + 16t2 + -16t2 Combine like terms: 6 + -6 = 0 39 + -55t + -16t2 = 0 + 55t + -55t + 16t2 + -16t2 39 + -55t + -16t2 = 55t + -55t + 16t2 + -16t2 Combine like terms: 55t + -55t = 0 39 + -55t + -16t2 = 0 + 16t2 + -16t2 39 + -55t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 39 + -55t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -2.4375 + 3.4375t + t2 = 0 Move the constant term to the right: Add '2.4375' to each side of the equation. -2.4375 + 3.4375t + 2.4375 + t2 = 0 + 2.4375 Reorder the terms: -2.4375 + 2.4375 + 3.4375t + t2 = 0 + 2.4375 Combine like terms: -2.4375 + 2.4375 = 0.0000 0.0000 + 3.4375t + t2 = 0 + 2.4375 3.4375t + t2 = 0 + 2.4375 Combine like terms: 0 + 2.4375 = 2.4375 3.4375t + t2 = 2.4375 The t term is 3.4375t. Take half its coefficient (1.71875). Square it (2.954101563) and add it to both sides. Add '2.954101563' to each side of the equation. 3.4375t + 2.954101563 + t2 = 2.4375 + 2.954101563 Reorder the terms: 2.954101563 + 3.4375t + t2 = 2.4375 + 2.954101563 Combine like terms: 2.4375 + 2.954101563 = 5.391601563 2.954101563 + 3.4375t + t2 = 5.391601563 Factor a perfect square on the left side: (t + 1.71875)(t + 1.71875) = 5.391601563 Calculate the square root of the right side: 2.321982249 Break this problem into two subproblems by setting (t + 1.71875) equal to 2.321982249 and -2.321982249.Subproblem 1
t + 1.71875 = 2.321982249 Simplifying t + 1.71875 = 2.321982249 Reorder the terms: 1.71875 + t = 2.321982249 Solving 1.71875 + t = 2.321982249 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.71875' to each side of the equation. 1.71875 + -1.71875 + t = 2.321982249 + -1.71875 Combine like terms: 1.71875 + -1.71875 = 0.00000 0.00000 + t = 2.321982249 + -1.71875 t = 2.321982249 + -1.71875 Combine like terms: 2.321982249 + -1.71875 = 0.603232249 t = 0.603232249 Simplifying t = 0.603232249Subproblem 2
t + 1.71875 = -2.321982249 Simplifying t + 1.71875 = -2.321982249 Reorder the terms: 1.71875 + t = -2.321982249 Solving 1.71875 + t = -2.321982249 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.71875' to each side of the equation. 1.71875 + -1.71875 + t = -2.321982249 + -1.71875 Combine like terms: 1.71875 + -1.71875 = 0.00000 0.00000 + t = -2.321982249 + -1.71875 t = -2.321982249 + -1.71875 Combine like terms: -2.321982249 + -1.71875 = -4.040732249 t = -4.040732249 Simplifying t = -4.040732249Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.603232249, -4.040732249}
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