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Simplifying 100 = -16t2 + 192t + 30 Reorder the terms: 100 = 30 + 192t + -16t2 Solving 100 = 30 + 192t + -16t2 Solving for variable 't'. Combine like terms: 100 + -30 = 70 70 + -192t + 16t2 = 30 + 192t + -16t2 + -30 + -192t + 16t2 Reorder the terms: 70 + -192t + 16t2 = 30 + -30 + 192t + -192t + -16t2 + 16t2 Combine like terms: 30 + -30 = 0 70 + -192t + 16t2 = 0 + 192t + -192t + -16t2 + 16t2 70 + -192t + 16t2 = 192t + -192t + -16t2 + 16t2 Combine like terms: 192t + -192t = 0 70 + -192t + 16t2 = 0 + -16t2 + 16t2 70 + -192t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 70 + -192t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(35 + -96t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(35 + -96t + 8t2)' equal to zero and attempt to solve: Simplifying 35 + -96t + 8t2 = 0 Solving 35 + -96t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 4.375 + -12t + t2 = 0 Move the constant term to the right: Add '-4.375' to each side of the equation. 4.375 + -12t + -4.375 + t2 = 0 + -4.375 Reorder the terms: 4.375 + -4.375 + -12t + t2 = 0 + -4.375 Combine like terms: 4.375 + -4.375 = 0.000 0.000 + -12t + t2 = 0 + -4.375 -12t + t2 = 0 + -4.375 Combine like terms: 0 + -4.375 = -4.375 -12t + t2 = -4.375 The t term is -12t. Take half its coefficient (-6). Square it (36) and add it to both sides. Add '36' to each side of the equation. -12t + 36 + t2 = -4.375 + 36 Reorder the terms: 36 + -12t + t2 = -4.375 + 36 Combine like terms: -4.375 + 36 = 31.625 36 + -12t + t2 = 31.625 Factor a perfect square on the left side: (t + -6)(t + -6) = 31.625 Calculate the square root of the right side: 5.62361094 Break this problem into two subproblems by setting (t + -6) equal to 5.62361094 and -5.62361094.Subproblem 1
t + -6 = 5.62361094 Simplifying t + -6 = 5.62361094 Reorder the terms: -6 + t = 5.62361094 Solving -6 + t = 5.62361094 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + t = 5.62361094 + 6 Combine like terms: -6 + 6 = 0 0 + t = 5.62361094 + 6 t = 5.62361094 + 6 Combine like terms: 5.62361094 + 6 = 11.62361094 t = 11.62361094 Simplifying t = 11.62361094Subproblem 2
t + -6 = -5.62361094 Simplifying t + -6 = -5.62361094 Reorder the terms: -6 + t = -5.62361094 Solving -6 + t = -5.62361094 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + t = -5.62361094 + 6 Combine like terms: -6 + 6 = 0 0 + t = -5.62361094 + 6 t = -5.62361094 + 6 Combine like terms: -5.62361094 + 6 = 0.37638906 t = 0.37638906 Simplifying t = 0.37638906Solution
The solution to the problem is based on the solutions from the subproblems. t = {11.62361094, 0.37638906}Solution
t = {11.62361094, 0.37638906}
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