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Simplifying 0 = t2 + 90t + -120 Reorder the terms: 0 = -120 + 90t + t2 Solving 0 = -120 + 90t + t2 Solving for variable 't'. Combine like terms: 0 + 120 = 120 120 + -90t + -1t2 = -120 + 90t + t2 + 120 + -90t + -1t2 Reorder the terms: 120 + -90t + -1t2 = -120 + 120 + 90t + -90t + t2 + -1t2 Combine like terms: -120 + 120 = 0 120 + -90t + -1t2 = 0 + 90t + -90t + t2 + -1t2 120 + -90t + -1t2 = 90t + -90t + t2 + -1t2 Combine like terms: 90t + -90t = 0 120 + -90t + -1t2 = 0 + t2 + -1t2 120 + -90t + -1t2 = t2 + -1t2 Combine like terms: t2 + -1t2 = 0 120 + -90t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -120 + 90t + t2 = 0 Move the constant term to the right: Add '120' to each side of the equation. -120 + 90t + 120 + t2 = 0 + 120 Reorder the terms: -120 + 120 + 90t + t2 = 0 + 120 Combine like terms: -120 + 120 = 0 0 + 90t + t2 = 0 + 120 90t + t2 = 0 + 120 Combine like terms: 0 + 120 = 120 90t + t2 = 120 The t term is 90t. Take half its coefficient (45). Square it (2025) and add it to both sides. Add '2025' to each side of the equation. 90t + 2025 + t2 = 120 + 2025 Reorder the terms: 2025 + 90t + t2 = 120 + 2025 Combine like terms: 120 + 2025 = 2145 2025 + 90t + t2 = 2145 Factor a perfect square on the left side: (t + 45)(t + 45) = 2145 Calculate the square root of the right side: 46.314144708 Break this problem into two subproblems by setting (t + 45) equal to 46.314144708 and -46.314144708.Subproblem 1
t + 45 = 46.314144708 Simplifying t + 45 = 46.314144708 Reorder the terms: 45 + t = 46.314144708 Solving 45 + t = 46.314144708 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-45' to each side of the equation. 45 + -45 + t = 46.314144708 + -45 Combine like terms: 45 + -45 = 0 0 + t = 46.314144708 + -45 t = 46.314144708 + -45 Combine like terms: 46.314144708 + -45 = 1.314144708 t = 1.314144708 Simplifying t = 1.314144708Subproblem 2
t + 45 = -46.314144708 Simplifying t + 45 = -46.314144708 Reorder the terms: 45 + t = -46.314144708 Solving 45 + t = -46.314144708 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-45' to each side of the equation. 45 + -45 + t = -46.314144708 + -45 Combine like terms: 45 + -45 = 0 0 + t = -46.314144708 + -45 t = -46.314144708 + -45 Combine like terms: -46.314144708 + -45 = -91.314144708 t = -91.314144708 Simplifying t = -91.314144708Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.314144708, -91.314144708}
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