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Simplifying (2t + -3)(2t + -3) + -2 = 24 Reorder the terms: (-3 + 2t)(2t + -3) + -2 = 24 Reorder the terms: (-3 + 2t)(-3 + 2t) + -2 = 24 Multiply (-3 + 2t) * (-3 + 2t) (-3(-3 + 2t) + 2t * (-3 + 2t)) + -2 = 24 ((-3 * -3 + 2t * -3) + 2t * (-3 + 2t)) + -2 = 24 ((9 + -6t) + 2t * (-3 + 2t)) + -2 = 24 (9 + -6t + (-3 * 2t + 2t * 2t)) + -2 = 24 (9 + -6t + (-6t + 4t2)) + -2 = 24 Combine like terms: -6t + -6t = -12t (9 + -12t + 4t2) + -2 = 24 Reorder the terms: 9 + -2 + -12t + 4t2 = 24 Combine like terms: 9 + -2 = 7 7 + -12t + 4t2 = 24 Solving 7 + -12t + 4t2 = 24 Solving for variable 't'. Reorder the terms: 7 + -24 + -12t + 4t2 = 24 + -24 Combine like terms: 7 + -24 = -17 -17 + -12t + 4t2 = 24 + -24 Combine like terms: 24 + -24 = 0 -17 + -12t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -4.25 + -3t + t2 = 0 Move the constant term to the right: Add '4.25' to each side of the equation. -4.25 + -3t + 4.25 + t2 = 0 + 4.25 Reorder the terms: -4.25 + 4.25 + -3t + t2 = 0 + 4.25 Combine like terms: -4.25 + 4.25 = 0.00 0.00 + -3t + t2 = 0 + 4.25 -3t + t2 = 0 + 4.25 Combine like terms: 0 + 4.25 = 4.25 -3t + t2 = 4.25 The t term is -3t. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3t + 2.25 + t2 = 4.25 + 2.25 Reorder the terms: 2.25 + -3t + t2 = 4.25 + 2.25 Combine like terms: 4.25 + 2.25 = 6.5 2.25 + -3t + t2 = 6.5 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 6.5 Calculate the square root of the right side: 2.549509757 Break this problem into two subproblems by setting (t + -1.5) equal to 2.549509757 and -2.549509757.Subproblem 1
t + -1.5 = 2.549509757 Simplifying t + -1.5 = 2.549509757 Reorder the terms: -1.5 + t = 2.549509757 Solving -1.5 + t = 2.549509757 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = 2.549509757 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 2.549509757 + 1.5 t = 2.549509757 + 1.5 Combine like terms: 2.549509757 + 1.5 = 4.049509757 t = 4.049509757 Simplifying t = 4.049509757Subproblem 2
t + -1.5 = -2.549509757 Simplifying t + -1.5 = -2.549509757 Reorder the terms: -1.5 + t = -2.549509757 Solving -1.5 + t = -2.549509757 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = -2.549509757 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -2.549509757 + 1.5 t = -2.549509757 + 1.5 Combine like terms: -2.549509757 + 1.5 = -1.049509757 t = -1.049509757 Simplifying t = -1.049509757Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.049509757, -1.049509757}
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