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Simplifying 90 + 2t(t + 10.5) = 180 Reorder the terms: 90 + 2t(10.5 + t) = 180 90 + (10.5 * 2t + t * 2t) = 180 90 + (21t + 2t2) = 180 Solving 90 + 21t + 2t2 = 180 Solving for variable 't'. Reorder the terms: 90 + -180 + 21t + 2t2 = 180 + -180 Combine like terms: 90 + -180 = -90 -90 + 21t + 2t2 = 180 + -180 Combine like terms: 180 + -180 = 0 -90 + 21t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -45 + 10.5t + t2 = 0 Move the constant term to the right: Add '45' to each side of the equation. -45 + 10.5t + 45 + t2 = 0 + 45 Reorder the terms: -45 + 45 + 10.5t + t2 = 0 + 45 Combine like terms: -45 + 45 = 0 0 + 10.5t + t2 = 0 + 45 10.5t + t2 = 0 + 45 Combine like terms: 0 + 45 = 45 10.5t + t2 = 45 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 = 45 + 27.5625 Reorder the terms: 27.5625 + 10.5t + t2 = 45 + 27.5625 Combine like terms: 45 + 27.5625 = 72.5625 27.5625 + 10.5t + t2 = 72.5625 Factor a perfect square on the left side: (t + 5.25)(t + 5.25) = 72.5625 Calculate the square root of the right side: 8.518362519 Break this problem into two subproblems by setting (t + 5.25) equal to 8.518362519 and -8.518362519.Subproblem 1
t + 5.25 = 8.518362519 Simplifying t + 5.25 = 8.518362519 Reorder the terms: 5.25 + t = 8.518362519 Solving 5.25 + t = 8.518362519 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 = 8.518362519 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + t = 8.518362519 + -5.25 t = 8.518362519 + -5.25 Combine like terms: 8.518362519 + -5.25 = 3.268362519 t = 3.268362519 Simplifying t = 3.268362519Subproblem 2
t + 5.25 = -8.518362519 Simplifying t + 5.25 = -8.518362519 Reorder the terms: 5.25 + t = -8.518362519 Solving 5.25 + t = -8.518362519 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 = -8.518362519 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + t = -8.518362519 + -5.25 t = -8.518362519 + -5.25 Combine like terms: -8.518362519 + -5.25 = -13.768362519 t = -13.768362519 Simplifying t = -13.768362519Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.268362519, -13.768362519}
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