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Simplifying 0 = 12t2 + -100t + -180 Reorder the terms: 0 = -180 + -100t + 12t2 Solving 0 = -180 + -100t + 12t2 Solving for variable 't'. Combine like terms: 0 + 180 = 180 180 + 100t + -12t2 = -180 + -100t + 12t2 + 180 + 100t + -12t2 Reorder the terms: 180 + 100t + -12t2 = -180 + 180 + -100t + 100t + 12t2 + -12t2 Combine like terms: -180 + 180 = 0 180 + 100t + -12t2 = 0 + -100t + 100t + 12t2 + -12t2 180 + 100t + -12t2 = -100t + 100t + 12t2 + -12t2 Combine like terms: -100t + 100t = 0 180 + 100t + -12t2 = 0 + 12t2 + -12t2 180 + 100t + -12t2 = 12t2 + -12t2 Combine like terms: 12t2 + -12t2 = 0 180 + 100t + -12t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(45 + 25t + -3t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(45 + 25t + -3t2)' equal to zero and attempt to solve: Simplifying 45 + 25t + -3t2 = 0 Solving 45 + 25t + -3t2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -15 + -8.333333333t + t2 = 0 Move the constant term to the right: Add '15' to each side of the equation. -15 + -8.333333333t + 15 + t2 = 0 + 15 Reorder the terms: -15 + 15 + -8.333333333t + t2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + -8.333333333t + t2 = 0 + 15 -8.333333333t + t2 = 0 + 15 Combine like terms: 0 + 15 = 15 -8.333333333t + t2 = 15 The t term is -8.333333333t. Take half its coefficient (-4.166666667). Square it (17.36111111) and add it to both sides. Add '17.36111111' to each side of the equation. -8.333333333t + 17.36111111 + t2 = 15 + 17.36111111 Reorder the terms: 17.36111111 + -8.333333333t + t2 = 15 + 17.36111111 Combine like terms: 15 + 17.36111111 = 32.36111111 17.36111111 + -8.333333333t + t2 = 32.36111111 Factor a perfect square on the left side: (t + -4.166666667)(t + -4.166666667) = 32.36111111 Calculate the square root of the right side: 5.688682722 Break this problem into two subproblems by setting (t + -4.166666667) equal to 5.688682722 and -5.688682722.Subproblem 1
t + -4.166666667 = 5.688682722 Simplifying t + -4.166666667 = 5.688682722 Reorder the terms: -4.166666667 + t = 5.688682722 Solving -4.166666667 + t = 5.688682722 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.166666667' to each side of the equation. -4.166666667 + 4.166666667 + t = 5.688682722 + 4.166666667 Combine like terms: -4.166666667 + 4.166666667 = 0.000000000 0.000000000 + t = 5.688682722 + 4.166666667 t = 5.688682722 + 4.166666667 Combine like terms: 5.688682722 + 4.166666667 = 9.855349389 t = 9.855349389 Simplifying t = 9.855349389Subproblem 2
t + -4.166666667 = -5.688682722 Simplifying t + -4.166666667 = -5.688682722 Reorder the terms: -4.166666667 + t = -5.688682722 Solving -4.166666667 + t = -5.688682722 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.166666667' to each side of the equation. -4.166666667 + 4.166666667 + t = -5.688682722 + 4.166666667 Combine like terms: -4.166666667 + 4.166666667 = 0.000000000 0.000000000 + t = -5.688682722 + 4.166666667 t = -5.688682722 + 4.166666667 Combine like terms: -5.688682722 + 4.166666667 = -1.522016055 t = -1.522016055 Simplifying t = -1.522016055Solution
The solution to the problem is based on the solutions from the subproblems. t = {9.855349389, -1.522016055}Solution
t = {9.855349389, -1.522016055}
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