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Simplifying (t + 10.5) * 2t = 180 Reorder the terms: (10.5 + t) * 2t = 180 Reorder the terms for easier multiplication: 2t(10.5 + t) = 180 (10.5 * 2t + t * 2t) = 180 (21t + 2t2) = 180 Solving 21t + 2t2 = 180 Solving for variable 't'. Reorder the terms: -180 + 21t + 2t2 = 180 + -180 Combine like terms: 180 + -180 = 0 -180 + 21t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -90 + 10.5t + t2 = 0 Move the constant term to the right: Add '90' to each side of the equation. -90 + 10.5t + 90 + t2 = 0 + 90 Reorder the terms: -90 + 90 + 10.5t + t2 = 0 + 90 Combine like terms: -90 + 90 = 0 0 + 10.5t + t2 = 0 + 90 10.5t + t2 = 0 + 90 Combine like terms: 0 + 90 = 90 10.5t + t2 = 90 The t term is 10.5t. Take half its coefficient (5.25). Square it (27.5625) and add it to both sides. Add '27.5625' to each side of the equation. 10.5t + 27.5625 + t2 = 90 + 27.5625 Reorder the terms: 27.5625 + 10.5t + t2 = 90 + 27.5625 Combine like terms: 90 + 27.5625 = 117.5625 27.5625 + 10.5t + t2 = 117.5625 Factor a perfect square on the left side: (t + 5.25)(t + 5.25) = 117.5625 Calculate the square root of the right side: 10.842624221 Break this problem into two subproblems by setting (t + 5.25) equal to 10.842624221 and -10.842624221.Subproblem 1
t + 5.25 = 10.842624221 Simplifying t + 5.25 = 10.842624221 Reorder the terms: 5.25 + t = 10.842624221 Solving 5.25 + t = 10.842624221 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + t = 10.842624221 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + t = 10.842624221 + -5.25 t = 10.842624221 + -5.25 Combine like terms: 10.842624221 + -5.25 = 5.592624221 t = 5.592624221 Simplifying t = 5.592624221Subproblem 2
t + 5.25 = -10.842624221 Simplifying t + 5.25 = -10.842624221 Reorder the terms: 5.25 + t = -10.842624221 Solving 5.25 + t = -10.842624221 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + t = -10.842624221 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + t = -10.842624221 + -5.25 t = -10.842624221 + -5.25 Combine like terms: -10.842624221 + -5.25 = -16.092624221 t = -16.092624221 Simplifying t = -16.092624221Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.592624221, -16.092624221}
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