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Simplifying -15t2 + 135t + 10 = 0 Reorder the terms: 10 + 135t + -15t2 = 0 Solving 10 + 135t + -15t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '5'. 5(2 + 27t + -3t2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(2 + 27t + -3t2)' equal to zero and attempt to solve: Simplifying 2 + 27t + -3t2 = 0 Solving 2 + 27t + -3t2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -0.6666666667 + -9t + t2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + -9t + 0.6666666667 + t2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + -9t + t2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + -9t + t2 = 0 + 0.6666666667 -9t + t2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 -9t + t2 = 0.6666666667 The t term is -9t. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9t + 20.25 + t2 = 0.6666666667 + 20.25 Reorder the terms: 20.25 + -9t + t2 = 0.6666666667 + 20.25 Combine like terms: 0.6666666667 + 20.25 = 20.9166666667 20.25 + -9t + t2 = 20.9166666667 Factor a perfect square on the left side: (t + -4.5)(t + -4.5) = 20.9166666667 Calculate the square root of the right side: 4.573474245 Break this problem into two subproblems by setting (t + -4.5) equal to 4.573474245 and -4.573474245.Subproblem 1
t + -4.5 = 4.573474245 Simplifying t + -4.5 = 4.573474245 Reorder the terms: -4.5 + t = 4.573474245 Solving -4.5 + t = 4.573474245 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + t = 4.573474245 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = 4.573474245 + 4.5 t = 4.573474245 + 4.5 Combine like terms: 4.573474245 + 4.5 = 9.073474245 t = 9.073474245 Simplifying t = 9.073474245Subproblem 2
t + -4.5 = -4.573474245 Simplifying t + -4.5 = -4.573474245 Reorder the terms: -4.5 + t = -4.573474245 Solving -4.5 + t = -4.573474245 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + t = -4.573474245 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = -4.573474245 + 4.5 t = -4.573474245 + 4.5 Combine like terms: -4.573474245 + 4.5 = -0.073474245 t = -0.073474245 Simplifying t = -0.073474245Solution
The solution to the problem is based on the solutions from the subproblems. t = {9.073474245, -0.073474245}Solution
t = {9.073474245, -0.073474245}
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