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Simplifying -6t2 + 13t + 3 = 0 Reorder the terms: 3 + 13t + -6t2 = 0 Solving 3 + 13t + -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 + -2.166666667t + t2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -2.166666667t + 0.5 + t2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -2.166666667t + t2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -2.166666667t + t2 = 0 + 0.5 -2.166666667t + t2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -2.166666667t + t2 = 0.5 The t term is -2.166666667t. Take half its coefficient (-1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. -2.166666667t + 1.173611113 + t2 = 0.5 + 1.173611113 Reorder the terms: 1.173611113 + -2.166666667t + t2 = 0.5 + 1.173611113 Combine like terms: 0.5 + 1.173611113 = 1.673611113 1.173611113 + -2.166666667t + t2 = 1.673611113 Factor a perfect square on the left side: (t + -1.083333334)(t + -1.083333334) = 1.673611113 Calculate the square root of the right side: 1.293681225 Break this problem into two subproblems by setting (t + -1.083333334) equal to 1.293681225 and -1.293681225.Subproblem 1
t + -1.083333334 = 1.293681225 Simplifying t + -1.083333334 = 1.293681225 Reorder the terms: -1.083333334 + t = 1.293681225 Solving -1.083333334 + t = 1.293681225 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + t = 1.293681225 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + t = 1.293681225 + 1.083333334 t = 1.293681225 + 1.083333334 Combine like terms: 1.293681225 + 1.083333334 = 2.377014559 t = 2.377014559 Simplifying t = 2.377014559Subproblem 2
t + -1.083333334 = -1.293681225 Simplifying t + -1.083333334 = -1.293681225 Reorder the terms: -1.083333334 + t = -1.293681225 Solving -1.083333334 + t = -1.293681225 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + t = -1.293681225 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + t = -1.293681225 + 1.083333334 t = -1.293681225 + 1.083333334 Combine like terms: -1.293681225 + 1.083333334 = -0.210347891 t = -0.210347891 Simplifying t = -0.210347891Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.377014559, -0.210347891}
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