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Simplifying 180 = 13t2 + -100t Reorder the terms: 180 = -100t + 13t2 Solving 180 = -100t + 13t2 Solving for variable 't'. Reorder the terms: 180 + 100t + -13t2 = -100t + 100t + 13t2 + -13t2 Combine like terms: -100t + 100t = 0 180 + 100t + -13t2 = 0 + 13t2 + -13t2 180 + 100t + -13t2 = 13t2 + -13t2 Combine like terms: 13t2 + -13t2 = 0 180 + 100t + -13t2 = 0 Begin completing the square. Divide all terms by -13 the coefficient of the squared term: Divide each side by '-13'. -13.84615385 + -7.692307692t + t2 = 0 Move the constant term to the right: Add '13.84615385' to each side of the equation. -13.84615385 + -7.692307692t + 13.84615385 + t2 = 0 + 13.84615385 Reorder the terms: -13.84615385 + 13.84615385 + -7.692307692t + t2 = 0 + 13.84615385 Combine like terms: -13.84615385 + 13.84615385 = 0.00000000 0.00000000 + -7.692307692t + t2 = 0 + 13.84615385 -7.692307692t + t2 = 0 + 13.84615385 Combine like terms: 0 + 13.84615385 = 13.84615385 -7.692307692t + t2 = 13.84615385 The t term is -7.692307692t. Take half its coefficient (-3.846153846). Square it (14.79289941) and add it to both sides. Add '14.79289941' to each side of the equation. -7.692307692t + 14.79289941 + t2 = 13.84615385 + 14.79289941 Reorder the terms: 14.79289941 + -7.692307692t + t2 = 13.84615385 + 14.79289941 Combine like terms: 13.84615385 + 14.79289941 = 28.63905326 14.79289941 + -7.692307692t + t2 = 28.63905326 Factor a perfect square on the left side: (t + -3.846153846)(t + -3.846153846) = 28.63905326 Calculate the square root of the right side: 5.35154681 Break this problem into two subproblems by setting (t + -3.846153846) equal to 5.35154681 and -5.35154681.Subproblem 1
t + -3.846153846 = 5.35154681 Simplifying t + -3.846153846 = 5.35154681 Reorder the terms: -3.846153846 + t = 5.35154681 Solving -3.846153846 + t = 5.35154681 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.846153846' to each side of the equation. -3.846153846 + 3.846153846 + t = 5.35154681 + 3.846153846 Combine like terms: -3.846153846 + 3.846153846 = 0.000000000 0.000000000 + t = 5.35154681 + 3.846153846 t = 5.35154681 + 3.846153846 Combine like terms: 5.35154681 + 3.846153846 = 9.197700656 t = 9.197700656 Simplifying t = 9.197700656Subproblem 2
t + -3.846153846 = -5.35154681 Simplifying t + -3.846153846 = -5.35154681 Reorder the terms: -3.846153846 + t = -5.35154681 Solving -3.846153846 + t = -5.35154681 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.846153846' to each side of the equation. -3.846153846 + 3.846153846 + t = -5.35154681 + 3.846153846 Combine like terms: -3.846153846 + 3.846153846 = 0.000000000 0.000000000 + t = -5.35154681 + 3.846153846 t = -5.35154681 + 3.846153846 Combine like terms: -5.35154681 + 3.846153846 = -1.505392964 t = -1.505392964 Simplifying t = -1.505392964Solution
The solution to the problem is based on the solutions from the subproblems. t = {9.197700656, -1.505392964}
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