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Simplifying -2t(t + -3) + 8t = 9t + -8 Reorder the terms: -2t(-3 + t) + 8t = 9t + -8 (-3 * -2t + t * -2t) + 8t = 9t + -8 (6t + -2t2) + 8t = 9t + -8 Reorder the terms: 6t + 8t + -2t2 = 9t + -8 Combine like terms: 6t + 8t = 14t 14t + -2t2 = 9t + -8 Reorder the terms: 14t + -2t2 = -8 + 9t Solving 14t + -2t2 = -8 + 9t Solving for variable 't'. Reorder the terms: 8 + 14t + -9t + -2t2 = -8 + 9t + 8 + -9t Combine like terms: 14t + -9t = 5t 8 + 5t + -2t2 = -8 + 9t + 8 + -9t Reorder the terms: 8 + 5t + -2t2 = -8 + 8 + 9t + -9t Combine like terms: -8 + 8 = 0 8 + 5t + -2t2 = 0 + 9t + -9t 8 + 5t + -2t2 = 9t + -9t Combine like terms: 9t + -9t = 0 8 + 5t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -4 + -2.5t + t2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + -2.5t + 4 + t2 = 0 + 4 Reorder the terms: -4 + 4 + -2.5t + t2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -2.5t + t2 = 0 + 4 -2.5t + t2 = 0 + 4 Combine like terms: 0 + 4 = 4 -2.5t + t2 = 4 The t term is -2.5t. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5t + 1.5625 + t2 = 4 + 1.5625 Reorder the terms: 1.5625 + -2.5t + t2 = 4 + 1.5625 Combine like terms: 4 + 1.5625 = 5.5625 1.5625 + -2.5t + t2 = 5.5625 Factor a perfect square on the left side: (t + -1.25)(t + -1.25) = 5.5625 Calculate the square root of the right side: 2.358495283 Break this problem into two subproblems by setting (t + -1.25) equal to 2.358495283 and -2.358495283.Subproblem 1
t + -1.25 = 2.358495283 Simplifying t + -1.25 = 2.358495283 Reorder the terms: -1.25 + t = 2.358495283 Solving -1.25 + t = 2.358495283 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = 2.358495283 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = 2.358495283 + 1.25 t = 2.358495283 + 1.25 Combine like terms: 2.358495283 + 1.25 = 3.608495283 t = 3.608495283 Simplifying t = 3.608495283Subproblem 2
t + -1.25 = -2.358495283 Simplifying t + -1.25 = -2.358495283 Reorder the terms: -1.25 + t = -2.358495283 Solving -1.25 + t = -2.358495283 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = -2.358495283 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = -2.358495283 + 1.25 t = -2.358495283 + 1.25 Combine like terms: -2.358495283 + 1.25 = -1.108495283 t = -1.108495283 Simplifying t = -1.108495283Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.608495283, -1.108495283}
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