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Simplifying 3 = t(-16t + 45) Reorder the terms: 3 = t(45 + -16t) 3 = (45 * t + -16t * t) 3 = (45t + -16t2) Solving 3 = 45t + -16t2 Solving for variable 't'. Reorder the terms: 3 + -45t + 16t2 = 45t + -45t + -16t2 + 16t2 Combine like terms: 45t + -45t = 0 3 + -45t + 16t2 = 0 + -16t2 + 16t2 3 + -45t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 3 + -45t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.1875 + -2.8125t + t2 = 0 Move the constant term to the right: Add '-0.1875' to each side of the equation. 0.1875 + -2.8125t + -0.1875 + t2 = 0 + -0.1875 Reorder the terms: 0.1875 + -0.1875 + -2.8125t + t2 = 0 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + -2.8125t + t2 = 0 + -0.1875 -2.8125t + t2 = 0 + -0.1875 Combine like terms: 0 + -0.1875 = -0.1875 -2.8125t + t2 = -0.1875 The t term is -2.8125t. Take half its coefficient (-1.40625). Square it (1.977539063) and add it to both sides. Add '1.977539063' to each side of the equation. -2.8125t + 1.977539063 + t2 = -0.1875 + 1.977539063 Reorder the terms: 1.977539063 + -2.8125t + t2 = -0.1875 + 1.977539063 Combine like terms: -0.1875 + 1.977539063 = 1.790039063 1.977539063 + -2.8125t + t2 = 1.790039063 Factor a perfect square on the left side: (t + -1.40625)(t + -1.40625) = 1.790039063 Calculate the square root of the right side: 1.337923414 Break this problem into two subproblems by setting (t + -1.40625) equal to 1.337923414 and -1.337923414.Subproblem 1
t + -1.40625 = 1.337923414 Simplifying t + -1.40625 = 1.337923414 Reorder the terms: -1.40625 + t = 1.337923414 Solving -1.40625 + t = 1.337923414 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.40625' to each side of the equation. -1.40625 + 1.40625 + t = 1.337923414 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + t = 1.337923414 + 1.40625 t = 1.337923414 + 1.40625 Combine like terms: 1.337923414 + 1.40625 = 2.744173414 t = 2.744173414 Simplifying t = 2.744173414Subproblem 2
t + -1.40625 = -1.337923414 Simplifying t + -1.40625 = -1.337923414 Reorder the terms: -1.40625 + t = -1.337923414 Solving -1.40625 + t = -1.337923414 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.40625' to each side of the equation. -1.40625 + 1.40625 + t = -1.337923414 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + t = -1.337923414 + 1.40625 t = -1.337923414 + 1.40625 Combine like terms: -1.337923414 + 1.40625 = 0.068326586 t = 0.068326586 Simplifying t = 0.068326586Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.744173414, 0.068326586}
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