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Simplifying 7.5 = -16t2 + 120t Reorder the terms: 7.5 = 120t + -16t2 Solving 7.5 = 120t + -16t2 Solving for variable 't'. Reorder the terms: 7.5 + -120t + 16t2 = 120t + -120t + -16t2 + 16t2 Combine like terms: 120t + -120t = 0 7.5 + -120t + 16t2 = 0 + -16t2 + 16t2 7.5 + -120t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 7.5 + -120t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.46875 + -7.5t + t2 = 0 Move the constant term to the right: Add '-0.46875' to each side of the equation. 0.46875 + -7.5t + -0.46875 + t2 = 0 + -0.46875 Reorder the terms: 0.46875 + -0.46875 + -7.5t + t2 = 0 + -0.46875 Combine like terms: 0.46875 + -0.46875 = 0.00000 0.00000 + -7.5t + t2 = 0 + -0.46875 -7.5t + t2 = 0 + -0.46875 Combine like terms: 0 + -0.46875 = -0.46875 -7.5t + t2 = -0.46875 The t term is -7.5t. Take half its coefficient (-3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. -7.5t + 14.0625 + t2 = -0.46875 + 14.0625 Reorder the terms: 14.0625 + -7.5t + t2 = -0.46875 + 14.0625 Combine like terms: -0.46875 + 14.0625 = 13.59375 14.0625 + -7.5t + t2 = 13.59375 Factor a perfect square on the left side: (t + -3.75)(t + -3.75) = 13.59375 Calculate the square root of the right side: 3.686970301 Break this problem into two subproblems by setting (t + -3.75) equal to 3.686970301 and -3.686970301.Subproblem 1
t + -3.75 = 3.686970301 Simplifying t + -3.75 = 3.686970301 Reorder the terms: -3.75 + t = 3.686970301 Solving -3.75 + t = 3.686970301 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + t = 3.686970301 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + t = 3.686970301 + 3.75 t = 3.686970301 + 3.75 Combine like terms: 3.686970301 + 3.75 = 7.436970301 t = 7.436970301 Simplifying t = 7.436970301Subproblem 2
t + -3.75 = -3.686970301 Simplifying t + -3.75 = -3.686970301 Reorder the terms: -3.75 + t = -3.686970301 Solving -3.75 + t = -3.686970301 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + t = -3.686970301 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + t = -3.686970301 + 3.75 t = -3.686970301 + 3.75 Combine like terms: -3.686970301 + 3.75 = 0.063029699 t = 0.063029699 Simplifying t = 0.063029699Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.436970301, 0.063029699}
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