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Simplifying 3t2 + 10t + -12 = 0 Reorder the terms: -12 + 10t + 3t2 = 0 Solving -12 + 10t + 3t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -4 + 3.333333333t + t2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 3.333333333t + 4 + t2 = 0 + 4 Reorder the terms: -4 + 4 + 3.333333333t + t2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 3.333333333t + t2 = 0 + 4 3.333333333t + t2 = 0 + 4 Combine like terms: 0 + 4 = 4 3.333333333t + t2 = 4 The t term is 3.333333333t. Take half its coefficient (1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. 3.333333333t + 2.777777779 + t2 = 4 + 2.777777779 Reorder the terms: 2.777777779 + 3.333333333t + t2 = 4 + 2.777777779 Combine like terms: 4 + 2.777777779 = 6.777777779 2.777777779 + 3.333333333t + t2 = 6.777777779 Factor a perfect square on the left side: (t + 1.666666667)(t + 1.666666667) = 6.777777779 Calculate the square root of the right side: 2.603416559 Break this problem into two subproblems by setting (t + 1.666666667) equal to 2.603416559 and -2.603416559.Subproblem 1
t + 1.666666667 = 2.603416559 Simplifying t + 1.666666667 = 2.603416559 Reorder the terms: 1.666666667 + t = 2.603416559 Solving 1.666666667 + t = 2.603416559 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + t = 2.603416559 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + t = 2.603416559 + -1.666666667 t = 2.603416559 + -1.666666667 Combine like terms: 2.603416559 + -1.666666667 = 0.936749892 t = 0.936749892 Simplifying t = 0.936749892Subproblem 2
t + 1.666666667 = -2.603416559 Simplifying t + 1.666666667 = -2.603416559 Reorder the terms: 1.666666667 + t = -2.603416559 Solving 1.666666667 + t = -2.603416559 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + t = -2.603416559 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + t = -2.603416559 + -1.666666667 t = -2.603416559 + -1.666666667 Combine like terms: -2.603416559 + -1.666666667 = -4.270083226 t = -4.270083226 Simplifying t = -4.270083226Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.936749892, -4.270083226}
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