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Simplifying 6 = -4.9t2 + 8.1t + 3 Reorder the terms: 6 = 3 + 8.1t + -4.9t2 Solving 6 = 3 + 8.1t + -4.9t2 Solving for variable 't'. Combine like terms: 6 + -3 = 3 3 + -8.1t + 4.9t2 = 3 + 8.1t + -4.9t2 + -3 + -8.1t + 4.9t2 Reorder the terms: 3 + -8.1t + 4.9t2 = 3 + -3 + 8.1t + -8.1t + -4.9t2 + 4.9t2 Combine like terms: 3 + -3 = 0 3 + -8.1t + 4.9t2 = 0 + 8.1t + -8.1t + -4.9t2 + 4.9t2 3 + -8.1t + 4.9t2 = 8.1t + -8.1t + -4.9t2 + 4.9t2 Combine like terms: 8.1t + -8.1t = 0.0 3 + -8.1t + 4.9t2 = 0.0 + -4.9t2 + 4.9t2 3 + -8.1t + 4.9t2 = -4.9t2 + 4.9t2 Combine like terms: -4.9t2 + 4.9t2 = 0.0 3 + -8.1t + 4.9t2 = 0.0 Begin completing the square. Divide all terms by 4.9 the coefficient of the squared term: Divide each side by '4.9'. 0.612244898 + -1.653061224t + t2 = 0 Move the constant term to the right: Add '-0.612244898' to each side of the equation. 0.612244898 + -1.653061224t + -0.612244898 + t2 = 0 + -0.612244898 Reorder the terms: 0.612244898 + -0.612244898 + -1.653061224t + t2 = 0 + -0.612244898 Combine like terms: 0.612244898 + -0.612244898 = 0.000000000 0.000000000 + -1.653061224t + t2 = 0 + -0.612244898 -1.653061224t + t2 = 0 + -0.612244898 Combine like terms: 0 + -0.612244898 = -0.612244898 -1.653061224t + t2 = -0.612244898 The t term is -1.653061224t. Take half its coefficient (-0.826530612). Square it (0.6831528526) and add it to both sides. Add '0.6831528526' to each side of the equation. -1.653061224t + 0.6831528526 + t2 = -0.612244898 + 0.6831528526 Reorder the terms: 0.6831528526 + -1.653061224t + t2 = -0.612244898 + 0.6831528526 Combine like terms: -0.612244898 + 0.6831528526 = 0.0709079546 0.6831528526 + -1.653061224t + t2 = 0.0709079546 Factor a perfect square on the left side: (t + -0.826530612)(t + -0.826530612) = 0.0709079546 Calculate the square root of the right side: 0.266285476 Break this problem into two subproblems by setting (t + -0.826530612) equal to 0.266285476 and -0.266285476.Subproblem 1
t + -0.826530612 = 0.266285476 Simplifying t + -0.826530612 = 0.266285476 Reorder the terms: -0.826530612 + t = 0.266285476 Solving -0.826530612 + t = 0.266285476 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.826530612' to each side of the equation. -0.826530612 + 0.826530612 + t = 0.266285476 + 0.826530612 Combine like terms: -0.826530612 + 0.826530612 = 0.000000000 0.000000000 + t = 0.266285476 + 0.826530612 t = 0.266285476 + 0.826530612 Combine like terms: 0.266285476 + 0.826530612 = 1.092816088 t = 1.092816088 Simplifying t = 1.092816088Subproblem 2
t + -0.826530612 = -0.266285476 Simplifying t + -0.826530612 = -0.266285476 Reorder the terms: -0.826530612 + t = -0.266285476 Solving -0.826530612 + t = -0.266285476 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.826530612' to each side of the equation. -0.826530612 + 0.826530612 + t = -0.266285476 + 0.826530612 Combine like terms: -0.826530612 + 0.826530612 = 0.000000000 0.000000000 + t = -0.266285476 + 0.826530612 t = -0.266285476 + 0.826530612 Combine like terms: -0.266285476 + 0.826530612 = 0.560245136 t = 0.560245136 Simplifying t = 0.560245136Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.092816088, 0.560245136}
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