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Simplifying 128 = -16t2 + 128t + 2 Reorder the terms: 128 = 2 + 128t + -16t2 Solving 128 = 2 + 128t + -16t2 Solving for variable 't'. Combine like terms: 128 + -2 = 126 126 + -128t + 16t2 = 2 + 128t + -16t2 + -2 + -128t + 16t2 Reorder the terms: 126 + -128t + 16t2 = 2 + -2 + 128t + -128t + -16t2 + 16t2 Combine like terms: 2 + -2 = 0 126 + -128t + 16t2 = 0 + 128t + -128t + -16t2 + 16t2 126 + -128t + 16t2 = 128t + -128t + -16t2 + 16t2 Combine like terms: 128t + -128t = 0 126 + -128t + 16t2 = 0 + -16t2 + 16t2 126 + -128t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 126 + -128t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(63 + -64t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(63 + -64t + 8t2)' equal to zero and attempt to solve: Simplifying 63 + -64t + 8t2 = 0 Solving 63 + -64t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 7.875 + -8t + t2 = 0 Move the constant term to the right: Add '-7.875' to each side of the equation. 7.875 + -8t + -7.875 + t2 = 0 + -7.875 Reorder the terms: 7.875 + -7.875 + -8t + t2 = 0 + -7.875 Combine like terms: 7.875 + -7.875 = 0.000 0.000 + -8t + t2 = 0 + -7.875 -8t + t2 = 0 + -7.875 Combine like terms: 0 + -7.875 = -7.875 -8t + t2 = -7.875 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 = -7.875 + 16 Reorder the terms: 16 + -8t + t2 = -7.875 + 16 Combine like terms: -7.875 + 16 = 8.125 16 + -8t + t2 = 8.125 Factor a perfect square on the left side: (t + -4)(t + -4) = 8.125 Calculate the square root of the right side: 2.850438563 Break this problem into two subproblems by setting (t + -4) equal to 2.850438563 and -2.850438563.Subproblem 1
t + -4 = 2.850438563 Simplifying t + -4 = 2.850438563 Reorder the terms: -4 + t = 2.850438563 Solving -4 + t = 2.850438563 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 = 2.850438563 + 4 Combine like terms: -4 + 4 = 0 0 + t = 2.850438563 + 4 t = 2.850438563 + 4 Combine like terms: 2.850438563 + 4 = 6.850438563 t = 6.850438563 Simplifying t = 6.850438563Subproblem 2
t + -4 = -2.850438563 Simplifying t + -4 = -2.850438563 Reorder the terms: -4 + t = -2.850438563 Solving -4 + t = -2.850438563 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 = -2.850438563 + 4 Combine like terms: -4 + 4 = 0 0 + t = -2.850438563 + 4 t = -2.850438563 + 4 Combine like terms: -2.850438563 + 4 = 1.149561437 t = 1.149561437 Simplifying t = 1.149561437Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.850438563, 1.149561437}Solution
t = {6.850438563, 1.149561437}
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