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Simplifying 12000 = 12t2 + 324t + 3000 Reorder the terms: 12000 = 3000 + 324t + 12t2 Solving 12000 = 3000 + 324t + 12t2 Solving for variable 't'. Combine like terms: 12000 + -3000 = 9000 9000 + -324t + -12t2 = 3000 + 324t + 12t2 + -3000 + -324t + -12t2 Reorder the terms: 9000 + -324t + -12t2 = 3000 + -3000 + 324t + -324t + 12t2 + -12t2 Combine like terms: 3000 + -3000 = 0 9000 + -324t + -12t2 = 0 + 324t + -324t + 12t2 + -12t2 9000 + -324t + -12t2 = 324t + -324t + 12t2 + -12t2 Combine like terms: 324t + -324t = 0 9000 + -324t + -12t2 = 0 + 12t2 + -12t2 9000 + -324t + -12t2 = 12t2 + -12t2 Combine like terms: 12t2 + -12t2 = 0 9000 + -324t + -12t2 = 0 Factor out the Greatest Common Factor (GCF), '12'. 12(750 + -27t + -1t2) = 0 Ignore the factor 12.Subproblem 1
Set the factor '(750 + -27t + -1t2)' equal to zero and attempt to solve: Simplifying 750 + -27t + -1t2 = 0 Solving 750 + -27t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -750 + 27t + t2 = 0 Move the constant term to the right: Add '750' to each side of the equation. -750 + 27t + 750 + t2 = 0 + 750 Reorder the terms: -750 + 750 + 27t + t2 = 0 + 750 Combine like terms: -750 + 750 = 0 0 + 27t + t2 = 0 + 750 27t + t2 = 0 + 750 Combine like terms: 0 + 750 = 750 27t + t2 = 750 The t term is 27t. Take half its coefficient (13.5). Square it (182.25) and add it to both sides. Add '182.25' to each side of the equation. 27t + 182.25 + t2 = 750 + 182.25 Reorder the terms: 182.25 + 27t + t2 = 750 + 182.25 Combine like terms: 750 + 182.25 = 932.25 182.25 + 27t + t2 = 932.25 Factor a perfect square on the left side: (t + 13.5)(t + 13.5) = 932.25 Calculate the square root of the right side: 30.532769282 Break this problem into two subproblems by setting (t + 13.5) equal to 30.532769282 and -30.532769282.Subproblem 1
t + 13.5 = 30.532769282 Simplifying t + 13.5 = 30.532769282 Reorder the terms: 13.5 + t = 30.532769282 Solving 13.5 + t = 30.532769282 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-13.5' to each side of the equation. 13.5 + -13.5 + t = 30.532769282 + -13.5 Combine like terms: 13.5 + -13.5 = 0.0 0.0 + t = 30.532769282 + -13.5 t = 30.532769282 + -13.5 Combine like terms: 30.532769282 + -13.5 = 17.032769282 t = 17.032769282 Simplifying t = 17.032769282Subproblem 2
t + 13.5 = -30.532769282 Simplifying t + 13.5 = -30.532769282 Reorder the terms: 13.5 + t = -30.532769282 Solving 13.5 + t = -30.532769282 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-13.5' to each side of the equation. 13.5 + -13.5 + t = -30.532769282 + -13.5 Combine like terms: 13.5 + -13.5 = 0.0 0.0 + t = -30.532769282 + -13.5 t = -30.532769282 + -13.5 Combine like terms: -30.532769282 + -13.5 = -44.032769282 t = -44.032769282 Simplifying t = -44.032769282Solution
The solution to the problem is based on the solutions from the subproblems. t = {17.032769282, -44.032769282}Solution
t = {17.032769282, -44.032769282}
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