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Simplifying -3t2 + 18t + 20 = 2t + 5 Reorder the terms: 20 + 18t + -3t2 = 2t + 5 Reorder the terms: 20 + 18t + -3t2 = 5 + 2t Solving 20 + 18t + -3t2 = 5 + 2t Solving for variable 't'. Reorder the terms: 20 + -5 + 18t + -2t + -3t2 = 5 + 2t + -5 + -2t Combine like terms: 20 + -5 = 15 15 + 18t + -2t + -3t2 = 5 + 2t + -5 + -2t Combine like terms: 18t + -2t = 16t 15 + 16t + -3t2 = 5 + 2t + -5 + -2t Reorder the terms: 15 + 16t + -3t2 = 5 + -5 + 2t + -2t Combine like terms: 5 + -5 = 0 15 + 16t + -3t2 = 0 + 2t + -2t 15 + 16t + -3t2 = 2t + -2t Combine like terms: 2t + -2t = 0 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 = 12.111111113 7.111111113 + -5.333333333t + t2 = 12.111111113 Factor a perfect square on the left side: (t + -2.666666667)(t + -2.666666667) = 12.111111113 Calculate the square root of the right side: 3.48010217 Break this problem into two subproblems by setting (t + -2.666666667) equal to 3.48010217 and -3.48010217.Subproblem 1
t + -2.666666667 = 3.48010217 Simplifying t + -2.666666667 = 3.48010217 Reorder the terms: -2.666666667 + t = 3.48010217 Solving -2.666666667 + t = 3.48010217 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 = 3.48010217 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + t = 3.48010217 + 2.666666667 t = 3.48010217 + 2.666666667 Combine like terms: 3.48010217 + 2.666666667 = 6.146768837 t = 6.146768837 Simplifying t = 6.146768837Subproblem 2
t + -2.666666667 = -3.48010217 Simplifying t + -2.666666667 = -3.48010217 Reorder the terms: -2.666666667 + t = -3.48010217 Solving -2.666666667 + t = -3.48010217 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 = -3.48010217 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + t = -3.48010217 + 2.666666667 t = -3.48010217 + 2.666666667 Combine like terms: -3.48010217 + 2.666666667 = -0.813435503 t = -0.813435503 Simplifying t = -0.813435503Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.146768837, -0.813435503}
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