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Simplifying 30(3t2 + 2t + -30) = 0 Reorder the terms: 30(-30 + 2t + 3t2) = 0 (-30 * 30 + 2t * 30 + 3t2 * 30) = 0 (-900 + 60t + 90t2) = 0 Solving -900 + 60t + 90t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '30'. 30(-30 + 2t + 3t2) = 0 Ignore the factor 30.Subproblem 1
Set the factor '(-30 + 2t + 3t2)' equal to zero and attempt to solve: Simplifying -30 + 2t + 3t2 = 0 Solving -30 + 2t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -10 + 0.6666666667t + t2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + 0.6666666667t + 10 + t2 = 0 + 10 Reorder the terms: -10 + 10 + 0.6666666667t + t2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + 0.6666666667t + t2 = 0 + 10 0.6666666667t + t2 = 0 + 10 Combine like terms: 0 + 10 = 10 0.6666666667t + t2 = 10 The t term is 0.6666666667t. Take half its coefficient (0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. 0.6666666667t + 0.1111111112 + t2 = 10 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667t + t2 = 10 + 0.1111111112 Combine like terms: 10 + 0.1111111112 = 10.1111111112 0.1111111112 + 0.6666666667t + t2 = 10.1111111112 Factor a perfect square on the left side: (t + 0.3333333334)(t + 0.3333333334) = 10.1111111112 Calculate the square root of the right side: 3.179797338 Break this problem into two subproblems by setting (t + 0.3333333334) equal to 3.179797338 and -3.179797338.Subproblem 1
t + 0.3333333334 = 3.179797338 Simplifying t + 0.3333333334 = 3.179797338 Reorder the terms: 0.3333333334 + t = 3.179797338 Solving 0.3333333334 + t = 3.179797338 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + t = 3.179797338 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + t = 3.179797338 + -0.3333333334 t = 3.179797338 + -0.3333333334 Combine like terms: 3.179797338 + -0.3333333334 = 2.8464640046 t = 2.8464640046 Simplifying t = 2.8464640046Subproblem 2
t + 0.3333333334 = -3.179797338 Simplifying t + 0.3333333334 = -3.179797338 Reorder the terms: 0.3333333334 + t = -3.179797338 Solving 0.3333333334 + t = -3.179797338 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + t = -3.179797338 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + t = -3.179797338 + -0.3333333334 t = -3.179797338 + -0.3333333334 Combine like terms: -3.179797338 + -0.3333333334 = -3.5131306714 t = -3.5131306714 Simplifying t = -3.5131306714Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.8464640046, -3.5131306714}Solution
t = {2.8464640046, -3.5131306714}
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