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Simplifying 20 = -16t2 + 96t + 6 Reorder the terms: 20 = 6 + 96t + -16t2 Solving 20 = 6 + 96t + -16t2 Solving for variable 't'. Combine like terms: 20 + -6 = 14 14 + -96t + 16t2 = 6 + 96t + -16t2 + -6 + -96t + 16t2 Reorder the terms: 14 + -96t + 16t2 = 6 + -6 + 96t + -96t + -16t2 + 16t2 Combine like terms: 6 + -6 = 0 14 + -96t + 16t2 = 0 + 96t + -96t + -16t2 + 16t2 14 + -96t + 16t2 = 96t + -96t + -16t2 + 16t2 Combine like terms: 96t + -96t = 0 14 + -96t + 16t2 = 0 + -16t2 + 16t2 14 + -96t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 14 + -96t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(7 + -48t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(7 + -48t + 8t2)' equal to zero and attempt to solve: Simplifying 7 + -48t + 8t2 = 0 Solving 7 + -48t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.875 + -6t + t2 = 0 Move the constant term to the right: Add '-0.875' to each side of the equation. 0.875 + -6t + -0.875 + t2 = 0 + -0.875 Reorder the terms: 0.875 + -0.875 + -6t + t2 = 0 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + -6t + t2 = 0 + -0.875 -6t + t2 = 0 + -0.875 Combine like terms: 0 + -0.875 = -0.875 -6t + t2 = -0.875 The t term is -6t. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6t + 9 + t2 = -0.875 + 9 Reorder the terms: 9 + -6t + t2 = -0.875 + 9 Combine like terms: -0.875 + 9 = 8.125 9 + -6t + t2 = 8.125 Factor a perfect square on the left side: (t + -3)(t + -3) = 8.125 Calculate the square root of the right side: 2.850438563 Break this problem into two subproblems by setting (t + -3) equal to 2.850438563 and -2.850438563.Subproblem 1
t + -3 = 2.850438563 Simplifying t + -3 = 2.850438563 Reorder the terms: -3 + t = 2.850438563 Solving -3 + t = 2.850438563 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = 2.850438563 + 3 Combine like terms: -3 + 3 = 0 0 + t = 2.850438563 + 3 t = 2.850438563 + 3 Combine like terms: 2.850438563 + 3 = 5.850438563 t = 5.850438563 Simplifying t = 5.850438563Subproblem 2
t + -3 = -2.850438563 Simplifying t + -3 = -2.850438563 Reorder the terms: -3 + t = -2.850438563 Solving -3 + t = -2.850438563 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = -2.850438563 + 3 Combine like terms: -3 + 3 = 0 0 + t = -2.850438563 + 3 t = -2.850438563 + 3 Combine like terms: -2.850438563 + 3 = 0.149561437 t = 0.149561437 Simplifying t = 0.149561437Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.850438563, 0.149561437}Solution
t = {5.850438563, 0.149561437}
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