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Simplifying 24 = -16t2 + 45t + 9 Reorder the terms: 24 = 9 + 45t + -16t2 Solving 24 = 9 + 45t + -16t2 Solving for variable 't'. Combine like terms: 24 + -9 = 15 15 + -45t + 16t2 = 9 + 45t + -16t2 + -9 + -45t + 16t2 Reorder the terms: 15 + -45t + 16t2 = 9 + -9 + 45t + -45t + -16t2 + 16t2 Combine like terms: 9 + -9 = 0 15 + -45t + 16t2 = 0 + 45t + -45t + -16t2 + 16t2 15 + -45t + 16t2 = 45t + -45t + -16t2 + 16t2 Combine like terms: 45t + -45t = 0 15 + -45t + 16t2 = 0 + -16t2 + 16t2 15 + -45t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 15 + -45t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.9375 + -2.8125t + t2 = 0 Move the constant term to the right: Add '-0.9375' to each side of the equation. 0.9375 + -2.8125t + -0.9375 + t2 = 0 + -0.9375 Reorder the terms: 0.9375 + -0.9375 + -2.8125t + t2 = 0 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + -2.8125t + t2 = 0 + -0.9375 -2.8125t + t2 = 0 + -0.9375 Combine like terms: 0 + -0.9375 = -0.9375 -2.8125t + t2 = -0.9375 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 = -0.9375 + 1.977539063 Reorder the terms: 1.977539063 + -2.8125t + t2 = -0.9375 + 1.977539063 Combine like terms: -0.9375 + 1.977539063 = 1.040039063 1.977539063 + -2.8125t + t2 = 1.040039063 Factor a perfect square on the left side: (t + -1.40625)(t + -1.40625) = 1.040039063 Calculate the square root of the right side: 1.019823055 Break this problem into two subproblems by setting (t + -1.40625) equal to 1.019823055 and -1.019823055.Subproblem 1
t + -1.40625 = 1.019823055 Simplifying t + -1.40625 = 1.019823055 Reorder the terms: -1.40625 + t = 1.019823055 Solving -1.40625 + t = 1.019823055 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 = 1.019823055 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + t = 1.019823055 + 1.40625 t = 1.019823055 + 1.40625 Combine like terms: 1.019823055 + 1.40625 = 2.426073055 t = 2.426073055 Simplifying t = 2.426073055Subproblem 2
t + -1.40625 = -1.019823055 Simplifying t + -1.40625 = -1.019823055 Reorder the terms: -1.40625 + t = -1.019823055 Solving -1.40625 + t = -1.019823055 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 = -1.019823055 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + t = -1.019823055 + 1.40625 t = -1.019823055 + 1.40625 Combine like terms: -1.019823055 + 1.40625 = 0.386426945 t = 0.386426945 Simplifying t = 0.386426945Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.426073055, 0.386426945}
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