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Simplifying 3 = -16t2 + -15t + 30 Reorder the terms: 3 = 30 + -15t + -16t2 Solving 3 = 30 + -15t + -16t2 Solving for variable 't'. Combine like terms: 3 + -30 = -27 -27 + 15t + 16t2 = 30 + -15t + -16t2 + -30 + 15t + 16t2 Reorder the terms: -27 + 15t + 16t2 = 30 + -30 + -15t + 15t + -16t2 + 16t2 Combine like terms: 30 + -30 = 0 -27 + 15t + 16t2 = 0 + -15t + 15t + -16t2 + 16t2 -27 + 15t + 16t2 = -15t + 15t + -16t2 + 16t2 Combine like terms: -15t + 15t = 0 -27 + 15t + 16t2 = 0 + -16t2 + 16t2 -27 + 15t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -27 + 15t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -1.6875 + 0.9375t + t2 = 0 Move the constant term to the right: Add '1.6875' to each side of the equation. -1.6875 + 0.9375t + 1.6875 + t2 = 0 + 1.6875 Reorder the terms: -1.6875 + 1.6875 + 0.9375t + t2 = 0 + 1.6875 Combine like terms: -1.6875 + 1.6875 = 0.0000 0.0000 + 0.9375t + t2 = 0 + 1.6875 0.9375t + t2 = 0 + 1.6875 Combine like terms: 0 + 1.6875 = 1.6875 0.9375t + t2 = 1.6875 The t term is 0.9375t. Take half its coefficient (0.46875). Square it (0.2197265625) and add it to both sides. Add '0.2197265625' to each side of the equation. 0.9375t + 0.2197265625 + t2 = 1.6875 + 0.2197265625 Reorder the terms: 0.2197265625 + 0.9375t + t2 = 1.6875 + 0.2197265625 Combine like terms: 1.6875 + 0.2197265625 = 1.9072265625 0.2197265625 + 0.9375t + t2 = 1.9072265625 Factor a perfect square on the left side: (t + 0.46875)(t + 0.46875) = 1.9072265625 Calculate the square root of the right side: 1.381023737 Break this problem into two subproblems by setting (t + 0.46875) equal to 1.381023737 and -1.381023737.Subproblem 1
t + 0.46875 = 1.381023737 Simplifying t + 0.46875 = 1.381023737 Reorder the terms: 0.46875 + t = 1.381023737 Solving 0.46875 + t = 1.381023737 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.46875' to each side of the equation. 0.46875 + -0.46875 + t = 1.381023737 + -0.46875 Combine like terms: 0.46875 + -0.46875 = 0.00000 0.00000 + t = 1.381023737 + -0.46875 t = 1.381023737 + -0.46875 Combine like terms: 1.381023737 + -0.46875 = 0.912273737 t = 0.912273737 Simplifying t = 0.912273737Subproblem 2
t + 0.46875 = -1.381023737 Simplifying t + 0.46875 = -1.381023737 Reorder the terms: 0.46875 + t = -1.381023737 Solving 0.46875 + t = -1.381023737 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.46875' to each side of the equation. 0.46875 + -0.46875 + t = -1.381023737 + -0.46875 Combine like terms: 0.46875 + -0.46875 = 0.00000 0.00000 + t = -1.381023737 + -0.46875 t = -1.381023737 + -0.46875 Combine like terms: -1.381023737 + -0.46875 = -1.849773737 t = -1.849773737 Simplifying t = -1.849773737Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.912273737, -1.849773737}
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