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Simplifying 45 = t2 + -1t Reorder the terms: 45 = -1t + t2 Solving 45 = -1t + t2 Solving for variable 't'. Reorder the terms: 45 + t + -1t2 = -1t + t + t2 + -1t2 Combine like terms: -1t + t = 0 45 + t + -1t2 = 0 + t2 + -1t2 45 + t + -1t2 = t2 + -1t2 Combine like terms: t2 + -1t2 = 0 45 + t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -45 + -1t + t2 = 0 Move the constant term to the right: Add '45' to each side of the equation. -45 + -1t + 45 + t2 = 0 + 45 Reorder the terms: -45 + 45 + -1t + t2 = 0 + 45 Combine like terms: -45 + 45 = 0 0 + -1t + t2 = 0 + 45 -1t + t2 = 0 + 45 Combine like terms: 0 + 45 = 45 -1t + t2 = 45 The t term is t. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1t + 0.25 + t2 = 45 + 0.25 Reorder the terms: 0.25 + -1t + t2 = 45 + 0.25 Combine like terms: 45 + 0.25 = 45.25 0.25 + -1t + t2 = 45.25 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 45.25 Calculate the square root of the right side: 6.726812024 Break this problem into two subproblems by setting (t + 0.5) equal to 6.726812024 and -6.726812024.Subproblem 1
t + 0.5 = 6.726812024 Simplifying t + 0.5 = 6.726812024 Reorder the terms: 0.5 + t = 6.726812024 Solving 0.5 + t = 6.726812024 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = 6.726812024 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 6.726812024 + -0.5 t = 6.726812024 + -0.5 Combine like terms: 6.726812024 + -0.5 = 6.226812024 t = 6.226812024 Simplifying t = 6.226812024Subproblem 2
t + 0.5 = -6.726812024 Simplifying t + 0.5 = -6.726812024 Reorder the terms: 0.5 + t = -6.726812024 Solving 0.5 + t = -6.726812024 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = -6.726812024 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -6.726812024 + -0.5 t = -6.726812024 + -0.5 Combine like terms: -6.726812024 + -0.5 = -7.226812024 t = -7.226812024 Simplifying t = -7.226812024Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.226812024, -7.226812024}
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