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Simplifying (t + -3)(t + -6) = 6 Reorder the terms: (-3 + t)(t + -6) = 6 Reorder the terms: (-3 + t)(-6 + t) = 6 Multiply (-3 + t) * (-6 + t) (-3(-6 + t) + t(-6 + t)) = 6 ((-6 * -3 + t * -3) + t(-6 + t)) = 6 ((18 + -3t) + t(-6 + t)) = 6 (18 + -3t + (-6 * t + t * t)) = 6 (18 + -3t + (-6t + t2)) = 6 Combine like terms: -3t + -6t = -9t (18 + -9t + t2) = 6 Solving 18 + -9t + t2 = 6 Solving for variable 't'. Reorder the terms: 18 + -6 + -9t + t2 = 6 + -6 Combine like terms: 18 + -6 = 12 12 + -9t + t2 = 6 + -6 Combine like terms: 6 + -6 = 0 12 + -9t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-12' to each side of the equation. 12 + -9t + -12 + t2 = 0 + -12 Reorder the terms: 12 + -12 + -9t + t2 = 0 + -12 Combine like terms: 12 + -12 = 0 0 + -9t + t2 = 0 + -12 -9t + t2 = 0 + -12 Combine like terms: 0 + -12 = -12 -9t + t2 = -12 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 = -12 + 20.25 Reorder the terms: 20.25 + -9t + t2 = -12 + 20.25 Combine like terms: -12 + 20.25 = 8.25 20.25 + -9t + t2 = 8.25 Factor a perfect square on the left side: (t + -4.5)(t + -4.5) = 8.25 Calculate the square root of the right side: 2.872281323 Break this problem into two subproblems by setting (t + -4.5) equal to 2.872281323 and -2.872281323.Subproblem 1
t + -4.5 = 2.872281323 Simplifying t + -4.5 = 2.872281323 Reorder the terms: -4.5 + t = 2.872281323 Solving -4.5 + t = 2.872281323 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 = 2.872281323 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = 2.872281323 + 4.5 t = 2.872281323 + 4.5 Combine like terms: 2.872281323 + 4.5 = 7.372281323 t = 7.372281323 Simplifying t = 7.372281323Subproblem 2
t + -4.5 = -2.872281323 Simplifying t + -4.5 = -2.872281323 Reorder the terms: -4.5 + t = -2.872281323 Solving -4.5 + t = -2.872281323 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 = -2.872281323 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = -2.872281323 + 4.5 t = -2.872281323 + 4.5 Combine like terms: -2.872281323 + 4.5 = 1.627718677 t = 1.627718677 Simplifying t = 1.627718677Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.372281323, 1.627718677}
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