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Simplifying 3t2 + -2t + -1 = 5 Reorder the terms: -1 + -2t + 3t2 = 5 Solving -1 + -2t + 3t2 = 5 Solving for variable 't'. Reorder the terms: -1 + -5 + -2t + 3t2 = 5 + -5 Combine like terms: -1 + -5 = -6 -6 + -2t + 3t2 = 5 + -5 Combine like terms: 5 + -5 = 0 -6 + -2t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2 + -0.6666666667t + t2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -0.6666666667t + 2 + t2 = 0 + 2 Reorder the terms: -2 + 2 + -0.6666666667t + t2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -0.6666666667t + t2 = 0 + 2 -0.6666666667t + t2 = 0 + 2 Combine like terms: 0 + 2 = 2 -0.6666666667t + t2 = 2 The t term is -0.6666666667t. Take half its coefficient (-0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. -0.6666666667t + 0.1111111112 + t2 = 2 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667t + t2 = 2 + 0.1111111112 Combine like terms: 2 + 0.1111111112 = 2.1111111112 0.1111111112 + -0.6666666667t + t2 = 2.1111111112 Factor a perfect square on the left side: (t + -0.3333333334)(t + -0.3333333334) = 2.1111111112 Calculate the square root of the right side: 1.452966315 Break this problem into two subproblems by setting (t + -0.3333333334) equal to 1.452966315 and -1.452966315.Subproblem 1
t + -0.3333333334 = 1.452966315 Simplifying t + -0.3333333334 = 1.452966315 Reorder the terms: -0.3333333334 + t = 1.452966315 Solving -0.3333333334 + t = 1.452966315 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + t = 1.452966315 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + t = 1.452966315 + 0.3333333334 t = 1.452966315 + 0.3333333334 Combine like terms: 1.452966315 + 0.3333333334 = 1.7862996484 t = 1.7862996484 Simplifying t = 1.7862996484Subproblem 2
t + -0.3333333334 = -1.452966315 Simplifying t + -0.3333333334 = -1.452966315 Reorder the terms: -0.3333333334 + t = -1.452966315 Solving -0.3333333334 + t = -1.452966315 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + t = -1.452966315 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + t = -1.452966315 + 0.3333333334 t = -1.452966315 + 0.3333333334 Combine like terms: -1.452966315 + 0.3333333334 = -1.1196329816 t = -1.1196329816 Simplifying t = -1.1196329816Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.7862996484, -1.1196329816}
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