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Simplifying 0 = -16t2 + 128t + 210 Reorder the terms: 0 = 210 + 128t + -16t2 Solving 0 = 210 + 128t + -16t2 Solving for variable 't'. Combine like terms: 0 + -210 = -210 -210 + -128t + 16t2 = 210 + 128t + -16t2 + -210 + -128t + 16t2 Reorder the terms: -210 + -128t + 16t2 = 210 + -210 + 128t + -128t + -16t2 + 16t2 Combine like terms: 210 + -210 = 0 -210 + -128t + 16t2 = 0 + 128t + -128t + -16t2 + 16t2 -210 + -128t + 16t2 = 128t + -128t + -16t2 + 16t2 Combine like terms: 128t + -128t = 0 -210 + -128t + 16t2 = 0 + -16t2 + 16t2 -210 + -128t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -210 + -128t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-105 + -64t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-105 + -64t + 8t2)' equal to zero and attempt to solve: Simplifying -105 + -64t + 8t2 = 0 Solving -105 + -64t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -13.125 + -8t + t2 = 0 Move the constant term to the right: Add '13.125' to each side of the equation. -13.125 + -8t + 13.125 + t2 = 0 + 13.125 Reorder the terms: -13.125 + 13.125 + -8t + t2 = 0 + 13.125 Combine like terms: -13.125 + 13.125 = 0.000 0.000 + -8t + t2 = 0 + 13.125 -8t + t2 = 0 + 13.125 Combine like terms: 0 + 13.125 = 13.125 -8t + t2 = 13.125 The t term is -8t. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8t + 16 + t2 = 13.125 + 16 Reorder the terms: 16 + -8t + t2 = 13.125 + 16 Combine like terms: 13.125 + 16 = 29.125 16 + -8t + t2 = 29.125 Factor a perfect square on the left side: (t + -4)(t + -4) = 29.125 Calculate the square root of the right side: 5.396758286 Break this problem into two subproblems by setting (t + -4) equal to 5.396758286 and -5.396758286.Subproblem 1
t + -4 = 5.396758286 Simplifying t + -4 = 5.396758286 Reorder the terms: -4 + t = 5.396758286 Solving -4 + t = 5.396758286 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + t = 5.396758286 + 4 Combine like terms: -4 + 4 = 0 0 + t = 5.396758286 + 4 t = 5.396758286 + 4 Combine like terms: 5.396758286 + 4 = 9.396758286 t = 9.396758286 Simplifying t = 9.396758286Subproblem 2
t + -4 = -5.396758286 Simplifying t + -4 = -5.396758286 Reorder the terms: -4 + t = -5.396758286 Solving -4 + t = -5.396758286 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + t = -5.396758286 + 4 Combine like terms: -4 + 4 = 0 0 + t = -5.396758286 + 4 t = -5.396758286 + 4 Combine like terms: -5.396758286 + 4 = -1.396758286 t = -1.396758286 Simplifying t = -1.396758286Solution
The solution to the problem is based on the solutions from the subproblems. t = {9.396758286, -1.396758286}Solution
t = {9.396758286, -1.396758286}
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