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Simplifying 8t2 + -67t + 31 = 0 Reorder the terms: 31 + -67t + 8t2 = 0 Solving 31 + -67t + 8t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 3.875 + -8.375t + t2 = 0 Move the constant term to the right: Add '-3.875' to each side of the equation. 3.875 + -8.375t + -3.875 + t2 = 0 + -3.875 Reorder the terms: 3.875 + -3.875 + -8.375t + t2 = 0 + -3.875 Combine like terms: 3.875 + -3.875 = 0.000 0.000 + -8.375t + t2 = 0 + -3.875 -8.375t + t2 = 0 + -3.875 Combine like terms: 0 + -3.875 = -3.875 -8.375t + t2 = -3.875 The t term is -8.375t. Take half its coefficient (-4.1875). Square it (17.53515625) and add it to both sides. Add '17.53515625' to each side of the equation. -8.375t + 17.53515625 + t2 = -3.875 + 17.53515625 Reorder the terms: 17.53515625 + -8.375t + t2 = -3.875 + 17.53515625 Combine like terms: -3.875 + 17.53515625 = 13.66015625 17.53515625 + -8.375t + t2 = 13.66015625 Factor a perfect square on the left side: (t + -4.1875)(t + -4.1875) = 13.66015625 Calculate the square root of the right side: 3.69596486 Break this problem into two subproblems by setting (t + -4.1875) equal to 3.69596486 and -3.69596486.Subproblem 1
t + -4.1875 = 3.69596486 Simplifying t + -4.1875 = 3.69596486 Reorder the terms: -4.1875 + t = 3.69596486 Solving -4.1875 + t = 3.69596486 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.1875' to each side of the equation. -4.1875 + 4.1875 + t = 3.69596486 + 4.1875 Combine like terms: -4.1875 + 4.1875 = 0.0000 0.0000 + t = 3.69596486 + 4.1875 t = 3.69596486 + 4.1875 Combine like terms: 3.69596486 + 4.1875 = 7.88346486 t = 7.88346486 Simplifying t = 7.88346486Subproblem 2
t + -4.1875 = -3.69596486 Simplifying t + -4.1875 = -3.69596486 Reorder the terms: -4.1875 + t = -3.69596486 Solving -4.1875 + t = -3.69596486 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.1875' to each side of the equation. -4.1875 + 4.1875 + t = -3.69596486 + 4.1875 Combine like terms: -4.1875 + 4.1875 = 0.0000 0.0000 + t = -3.69596486 + 4.1875 t = -3.69596486 + 4.1875 Combine like terms: -3.69596486 + 4.1875 = 0.49153514 t = 0.49153514 Simplifying t = 0.49153514Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.88346486, 0.49153514}
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