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Simplifying t[-6 + -1(2t + 3)] = -24 Reorder the terms: t[-6 + -1(3 + 2t)] = -24 t[-6 + (3 * -1 + 2t * -1)] = -24 t[-6 + (-3 + -2t)] = -24 Combine like terms: -6 + -3 = -9 t[-9 + -2t] = -24 [-9 * t + -2t * t] = -24 [-9t + -2t2] = -24 Solving -9t + -2t2 = -24 Solving for variable 't'. Reorder the terms: 24 + -9t + -2t2 = -24 + 24 Combine like terms: -24 + 24 = 0 24 + -9t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -12 + 4.5t + t2 = 0 Move the constant term to the right: Add '12' to each side of the equation. -12 + 4.5t + 12 + t2 = 0 + 12 Reorder the terms: -12 + 12 + 4.5t + t2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + 4.5t + t2 = 0 + 12 4.5t + t2 = 0 + 12 Combine like terms: 0 + 12 = 12 4.5t + t2 = 12 The t term is 4.5t. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5t + 5.0625 + t2 = 12 + 5.0625 Reorder the terms: 5.0625 + 4.5t + t2 = 12 + 5.0625 Combine like terms: 12 + 5.0625 = 17.0625 5.0625 + 4.5t + t2 = 17.0625 Factor a perfect square on the left side: (t + 2.25)(t + 2.25) = 17.0625 Calculate the square root of the right side: 4.13067791 Break this problem into two subproblems by setting (t + 2.25) equal to 4.13067791 and -4.13067791.Subproblem 1
t + 2.25 = 4.13067791 Simplifying t + 2.25 = 4.13067791 Reorder the terms: 2.25 + t = 4.13067791 Solving 2.25 + t = 4.13067791 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + t = 4.13067791 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + t = 4.13067791 + -2.25 t = 4.13067791 + -2.25 Combine like terms: 4.13067791 + -2.25 = 1.88067791 t = 1.88067791 Simplifying t = 1.88067791Subproblem 2
t + 2.25 = -4.13067791 Simplifying t + 2.25 = -4.13067791 Reorder the terms: 2.25 + t = -4.13067791 Solving 2.25 + t = -4.13067791 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + t = -4.13067791 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + t = -4.13067791 + -2.25 t = -4.13067791 + -2.25 Combine like terms: -4.13067791 + -2.25 = -6.38067791 t = -6.38067791 Simplifying t = -6.38067791Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.88067791, -6.38067791}
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