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Simplifying 6 = 6t2 + 30t + 15 Reorder the terms: 6 = 15 + 30t + 6t2 Solving 6 = 15 + 30t + 6t2 Solving for variable 't'. Combine like terms: 6 + -15 = -9 -9 + -30t + -6t2 = 15 + 30t + 6t2 + -15 + -30t + -6t2 Reorder the terms: -9 + -30t + -6t2 = 15 + -15 + 30t + -30t + 6t2 + -6t2 Combine like terms: 15 + -15 = 0 -9 + -30t + -6t2 = 0 + 30t + -30t + 6t2 + -6t2 -9 + -30t + -6t2 = 30t + -30t + 6t2 + -6t2 Combine like terms: 30t + -30t = 0 -9 + -30t + -6t2 = 0 + 6t2 + -6t2 -9 + -30t + -6t2 = 6t2 + -6t2 Combine like terms: 6t2 + -6t2 = 0 -9 + -30t + -6t2 = 0 Factor out the Greatest Common Factor (GCF), '-3'. -3(3 + 10t + 2t2) = 0 Ignore the factor -3.Subproblem 1
Set the factor '(3 + 10t + 2t2)' equal to zero and attempt to solve: Simplifying 3 + 10t + 2t2 = 0 Solving 3 + 10t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1.5 + 5t + t2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + 5t + -1.5 + t2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + 5t + t2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + 5t + t2 = 0 + -1.5 5t + t2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 5t + t2 = -1.5 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 = -1.5 + 6.25 Reorder the terms: 6.25 + 5t + t2 = -1.5 + 6.25 Combine like terms: -1.5 + 6.25 = 4.75 6.25 + 5t + t2 = 4.75 Factor a perfect square on the left side: (t + 2.5)(t + 2.5) = 4.75 Calculate the square root of the right side: 2.179449472 Break this problem into two subproblems by setting (t + 2.5) equal to 2.179449472 and -2.179449472.Subproblem 1
t + 2.5 = 2.179449472 Simplifying t + 2.5 = 2.179449472 Reorder the terms: 2.5 + t = 2.179449472 Solving 2.5 + t = 2.179449472 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 = 2.179449472 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = 2.179449472 + -2.5 t = 2.179449472 + -2.5 Combine like terms: 2.179449472 + -2.5 = -0.320550528 t = -0.320550528 Simplifying t = -0.320550528Subproblem 2
t + 2.5 = -2.179449472 Simplifying t + 2.5 = -2.179449472 Reorder the terms: 2.5 + t = -2.179449472 Solving 2.5 + t = -2.179449472 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 = -2.179449472 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = -2.179449472 + -2.5 t = -2.179449472 + -2.5 Combine like terms: -2.179449472 + -2.5 = -4.679449472 t = -4.679449472 Simplifying t = -4.679449472Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.320550528, -4.679449472}Solution
t = {-0.320550528, -4.679449472}
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