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Simplifying 12t2 + -110t + -170 = 0 Reorder the terms: -170 + -110t + 12t2 = 0 Solving -170 + -110t + 12t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(-85 + -55t + 6t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-85 + -55t + 6t2)' equal to zero and attempt to solve: Simplifying -85 + -55t + 6t2 = 0 Solving -85 + -55t + 6t2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -14.16666667 + -9.166666667t + t2 = 0 Move the constant term to the right: Add '14.16666667' to each side of the equation. -14.16666667 + -9.166666667t + 14.16666667 + t2 = 0 + 14.16666667 Reorder the terms: -14.16666667 + 14.16666667 + -9.166666667t + t2 = 0 + 14.16666667 Combine like terms: -14.16666667 + 14.16666667 = 0.00000000 0.00000000 + -9.166666667t + t2 = 0 + 14.16666667 -9.166666667t + t2 = 0 + 14.16666667 Combine like terms: 0 + 14.16666667 = 14.16666667 -9.166666667t + t2 = 14.16666667 The t term is -9.166666667t. Take half its coefficient (-4.583333334). Square it (21.00694445) and add it to both sides. Add '21.00694445' to each side of the equation. -9.166666667t + 21.00694445 + t2 = 14.16666667 + 21.00694445 Reorder the terms: 21.00694445 + -9.166666667t + t2 = 14.16666667 + 21.00694445 Combine like terms: 14.16666667 + 21.00694445 = 35.17361112 21.00694445 + -9.166666667t + t2 = 35.17361112 Factor a perfect square on the left side: (t + -4.583333334)(t + -4.583333334) = 35.17361112 Calculate the square root of the right side: 5.93073445 Break this problem into two subproblems by setting (t + -4.583333334) equal to 5.93073445 and -5.93073445.Subproblem 1
t + -4.583333334 = 5.93073445 Simplifying t + -4.583333334 = 5.93073445 Reorder the terms: -4.583333334 + t = 5.93073445 Solving -4.583333334 + t = 5.93073445 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.583333334' to each side of the equation. -4.583333334 + 4.583333334 + t = 5.93073445 + 4.583333334 Combine like terms: -4.583333334 + 4.583333334 = 0.000000000 0.000000000 + t = 5.93073445 + 4.583333334 t = 5.93073445 + 4.583333334 Combine like terms: 5.93073445 + 4.583333334 = 10.514067784 t = 10.514067784 Simplifying t = 10.514067784Subproblem 2
t + -4.583333334 = -5.93073445 Simplifying t + -4.583333334 = -5.93073445 Reorder the terms: -4.583333334 + t = -5.93073445 Solving -4.583333334 + t = -5.93073445 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.583333334' to each side of the equation. -4.583333334 + 4.583333334 + t = -5.93073445 + 4.583333334 Combine like terms: -4.583333334 + 4.583333334 = 0.000000000 0.000000000 + t = -5.93073445 + 4.583333334 t = -5.93073445 + 4.583333334 Combine like terms: -5.93073445 + 4.583333334 = -1.347401116 t = -1.347401116 Simplifying t = -1.347401116Solution
The solution to the problem is based on the solutions from the subproblems. t = {10.514067784, -1.347401116}Solution
t = {10.514067784, -1.347401116}
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