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Simplifying 3t(9t3 + 30t + 45t2) = 0 Reorder the terms: 3t(30t + 45t2 + 9t3) = 0 (30t * 3t + 45t2 * 3t + 9t3 * 3t) = 0 (90t2 + 135t3 + 27t4) = 0 Solving 90t2 + 135t3 + 27t4 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '9t2'. 9t2(10 + 15t + 3t2) = 0 Ignore the factor 9.Subproblem 1
Set the factor 't2' equal to zero and attempt to solve: Simplifying t2 = 0 Solving t2 = 0 Move all terms containing t to the left, all other terms to the right. Simplifying t2 = 0 Take the square root of each side: t = {0}Subproblem 2
Set the factor '(10 + 15t + 3t2)' equal to zero and attempt to solve: Simplifying 10 + 15t + 3t2 = 0 Solving 10 + 15t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 3.333333333 + 5t + t2 = 0 Move the constant term to the right: Add '-3.333333333' to each side of the equation. 3.333333333 + 5t + -3.333333333 + t2 = 0 + -3.333333333 Reorder the terms: 3.333333333 + -3.333333333 + 5t + t2 = 0 + -3.333333333 Combine like terms: 3.333333333 + -3.333333333 = 0.000000000 0.000000000 + 5t + t2 = 0 + -3.333333333 5t + t2 = 0 + -3.333333333 Combine like terms: 0 + -3.333333333 = -3.333333333 5t + t2 = -3.333333333 The t term is 5t. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5t + 6.25 + t2 = -3.333333333 + 6.25 Reorder the terms: 6.25 + 5t + t2 = -3.333333333 + 6.25 Combine like terms: -3.333333333 + 6.25 = 2.916666667 6.25 + 5t + t2 = 2.916666667 Factor a perfect square on the left side: (t + 2.5)(t + 2.5) = 2.916666667 Calculate the square root of the right side: 1.707825128 Break this problem into two subproblems by setting (t + 2.5) equal to 1.707825128 and -1.707825128.Subproblem 1
t + 2.5 = 1.707825128 Simplifying t + 2.5 = 1.707825128 Reorder the terms: 2.5 + t = 1.707825128 Solving 2.5 + t = 1.707825128 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + t = 1.707825128 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = 1.707825128 + -2.5 t = 1.707825128 + -2.5 Combine like terms: 1.707825128 + -2.5 = -0.792174872 t = -0.792174872 Simplifying t = -0.792174872Subproblem 2
t + 2.5 = -1.707825128 Simplifying t + 2.5 = -1.707825128 Reorder the terms: 2.5 + t = -1.707825128 Solving 2.5 + t = -1.707825128 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + t = -1.707825128 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = -1.707825128 + -2.5 t = -1.707825128 + -2.5 Combine like terms: -1.707825128 + -2.5 = -4.207825128 t = -4.207825128 Simplifying t = -4.207825128Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.792174872, -4.207825128}Solution
t = {0, -0.792174872, -4.207825128}
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