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Simplifying 6t2 + -16t + -3 = 0 Reorder the terms: -3 + -16t + 6t2 = 0 Solving -3 + -16t + 6t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.5 + -2.666666667t + t2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -2.666666667t + 0.5 + t2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -2.666666667t + t2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -2.666666667t + t2 = 0 + 0.5 -2.666666667t + t2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -2.666666667t + t2 = 0.5 The t term is -2.666666667t. Take half its coefficient (-1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. -2.666666667t + 1.777777780 + t2 = 0.5 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667t + t2 = 0.5 + 1.777777780 Combine like terms: 0.5 + 1.777777780 = 2.27777778 1.777777780 + -2.666666667t + t2 = 2.27777778 Factor a perfect square on the left side: (t + -1.333333334)(t + -1.333333334) = 2.27777778 Calculate the square root of the right side: 1.509230857 Break this problem into two subproblems by setting (t + -1.333333334) equal to 1.509230857 and -1.509230857.Subproblem 1
t + -1.333333334 = 1.509230857 Simplifying t + -1.333333334 = 1.509230857 Reorder the terms: -1.333333334 + t = 1.509230857 Solving -1.333333334 + t = 1.509230857 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + t = 1.509230857 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + t = 1.509230857 + 1.333333334 t = 1.509230857 + 1.333333334 Combine like terms: 1.509230857 + 1.333333334 = 2.842564191 t = 2.842564191 Simplifying t = 2.842564191Subproblem 2
t + -1.333333334 = -1.509230857 Simplifying t + -1.333333334 = -1.509230857 Reorder the terms: -1.333333334 + t = -1.509230857 Solving -1.333333334 + t = -1.509230857 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + t = -1.509230857 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + t = -1.509230857 + 1.333333334 t = -1.509230857 + 1.333333334 Combine like terms: -1.509230857 + 1.333333334 = -0.175897523 t = -0.175897523 Simplifying t = -0.175897523Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.842564191, -0.175897523}
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