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Simplifying 0 = 3t2 + 16t + 15 Reorder the terms: 0 = 15 + 16t + 3t2 Solving 0 = 15 + 16t + 3t2 Solving for variable 't'. Combine like terms: 0 + -15 = -15 -15 + -16t + -3t2 = 15 + 16t + 3t2 + -15 + -16t + -3t2 Reorder the terms: -15 + -16t + -3t2 = 15 + -15 + 16t + -16t + 3t2 + -3t2 Combine like terms: 15 + -15 = 0 -15 + -16t + -3t2 = 0 + 16t + -16t + 3t2 + -3t2 -15 + -16t + -3t2 = 16t + -16t + 3t2 + -3t2 Combine like terms: 16t + -16t = 0 -15 + -16t + -3t2 = 0 + 3t2 + -3t2 -15 + -16t + -3t2 = 3t2 + -3t2 Combine like terms: 3t2 + -3t2 = 0 -15 + -16t + -3t2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(15 + 16t + 3t2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(15 + 16t + 3t2)' equal to zero and attempt to solve: Simplifying 15 + 16t + 3t2 = 0 Solving 15 + 16t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 5 + 5.333333333t + t2 = 0 Move the constant term to the right: Add '-5' to each side of the equation. 5 + 5.333333333t + -5 + t2 = 0 + -5 Reorder the terms: 5 + -5 + 5.333333333t + t2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 5.333333333t + t2 = 0 + -5 5.333333333t + t2 = 0 + -5 Combine like terms: 0 + -5 = -5 5.333333333t + t2 = -5 The t term is 5.333333333t. Take half its coefficient (2.666666667). Square it (7.111111113) and add it to both sides. Add '7.111111113' to each side of the equation. 5.333333333t + 7.111111113 + t2 = -5 + 7.111111113 Reorder the terms: 7.111111113 + 5.333333333t + t2 = -5 + 7.111111113 Combine like terms: -5 + 7.111111113 = 2.111111113 7.111111113 + 5.333333333t + t2 = 2.111111113 Factor a perfect square on the left side: (t + 2.666666667)(t + 2.666666667) = 2.111111113 Calculate the square root of the right side: 1.452966315 Break this problem into two subproblems by setting (t + 2.666666667) equal to 1.452966315 and -1.452966315.Subproblem 1
t + 2.666666667 = 1.452966315 Simplifying t + 2.666666667 = 1.452966315 Reorder the terms: 2.666666667 + t = 1.452966315 Solving 2.666666667 + t = 1.452966315 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + t = 1.452966315 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + t = 1.452966315 + -2.666666667 t = 1.452966315 + -2.666666667 Combine like terms: 1.452966315 + -2.666666667 = -1.213700352 t = -1.213700352 Simplifying t = -1.213700352Subproblem 2
t + 2.666666667 = -1.452966315 Simplifying t + 2.666666667 = -1.452966315 Reorder the terms: 2.666666667 + t = -1.452966315 Solving 2.666666667 + t = -1.452966315 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + t = -1.452966315 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + t = -1.452966315 + -2.666666667 t = -1.452966315 + -2.666666667 Combine like terms: -1.452966315 + -2.666666667 = -4.119632982 t = -4.119632982 Simplifying t = -4.119632982Solution
The solution to the problem is based on the solutions from the subproblems. t = {-1.213700352, -4.119632982}Solution
t = {-1.213700352, -4.119632982}
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