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Simplifying 0 = -3t2 + 30t + -23 Reorder the terms: 0 = -23 + 30t + -3t2 Solving 0 = -23 + 30t + -3t2 Solving for variable 't'. Combine like terms: 0 + 23 = 23 23 + -30t + 3t2 = -23 + 30t + -3t2 + 23 + -30t + 3t2 Reorder the terms: 23 + -30t + 3t2 = -23 + 23 + 30t + -30t + -3t2 + 3t2 Combine like terms: -23 + 23 = 0 23 + -30t + 3t2 = 0 + 30t + -30t + -3t2 + 3t2 23 + -30t + 3t2 = 30t + -30t + -3t2 + 3t2 Combine like terms: 30t + -30t = 0 23 + -30t + 3t2 = 0 + -3t2 + 3t2 23 + -30t + 3t2 = -3t2 + 3t2 Combine like terms: -3t2 + 3t2 = 0 23 + -30t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 7.666666667 + -10t + t2 = 0 Move the constant term to the right: Add '-7.666666667' to each side of the equation. 7.666666667 + -10t + -7.666666667 + t2 = 0 + -7.666666667 Reorder the terms: 7.666666667 + -7.666666667 + -10t + t2 = 0 + -7.666666667 Combine like terms: 7.666666667 + -7.666666667 = 0.000000000 0.000000000 + -10t + t2 = 0 + -7.666666667 -10t + t2 = 0 + -7.666666667 Combine like terms: 0 + -7.666666667 = -7.666666667 -10t + t2 = -7.666666667 The t term is -10t. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10t + 25 + t2 = -7.666666667 + 25 Reorder the terms: 25 + -10t + t2 = -7.666666667 + 25 Combine like terms: -7.666666667 + 25 = 17.333333333 25 + -10t + t2 = 17.333333333 Factor a perfect square on the left side: (t + -5)(t + -5) = 17.333333333 Calculate the square root of the right side: 4.163331999 Break this problem into two subproblems by setting (t + -5) equal to 4.163331999 and -4.163331999.Subproblem 1
t + -5 = 4.163331999 Simplifying t + -5 = 4.163331999 Reorder the terms: -5 + t = 4.163331999 Solving -5 + t = 4.163331999 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + t = 4.163331999 + 5 Combine like terms: -5 + 5 = 0 0 + t = 4.163331999 + 5 t = 4.163331999 + 5 Combine like terms: 4.163331999 + 5 = 9.163331999 t = 9.163331999 Simplifying t = 9.163331999Subproblem 2
t + -5 = -4.163331999 Simplifying t + -5 = -4.163331999 Reorder the terms: -5 + t = -4.163331999 Solving -5 + t = -4.163331999 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + t = -4.163331999 + 5 Combine like terms: -5 + 5 = 0 0 + t = -4.163331999 + 5 t = -4.163331999 + 5 Combine like terms: -4.163331999 + 5 = 0.836668001 t = 0.836668001 Simplifying t = 0.836668001Solution
The solution to the problem is based on the solutions from the subproblems. t = {9.163331999, 0.836668001}
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