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Simplifying t2 + 5 = -5t Reorder the terms: 5 + t2 = -5t Solving 5 + t2 = -5t Solving for variable 't'. Reorder the terms: 5 + 5t + t2 = -5t + 5t Combine like terms: -5t + 5t = 0 5 + 5t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-5' to each side of the equation. 5 + 5t + -5 + t2 = 0 + -5 Reorder the terms: 5 + -5 + 5t + t2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 5t + t2 = 0 + -5 5t + t2 = 0 + -5 Combine like terms: 0 + -5 = -5 5t + t2 = -5 The t term is 5t. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5t + 6.25 + t2 = -5 + 6.25 Reorder the terms: 6.25 + 5t + t2 = -5 + 6.25 Combine like terms: -5 + 6.25 = 1.25 6.25 + 5t + t2 = 1.25 Factor a perfect square on the left side: (t + 2.5)(t + 2.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (t + 2.5) equal to 1.118033989 and -1.118033989.Subproblem 1
t + 2.5 = 1.118033989 Simplifying t + 2.5 = 1.118033989 Reorder the terms: 2.5 + t = 1.118033989 Solving 2.5 + t = 1.118033989 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + t = 1.118033989 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = 1.118033989 + -2.5 t = 1.118033989 + -2.5 Combine like terms: 1.118033989 + -2.5 = -1.381966011 t = -1.381966011 Simplifying t = -1.381966011Subproblem 2
t + 2.5 = -1.118033989 Simplifying t + 2.5 = -1.118033989 Reorder the terms: 2.5 + t = -1.118033989 Solving 2.5 + t = -1.118033989 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + t = -1.118033989 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = -1.118033989 + -2.5 t = -1.118033989 + -2.5 Combine like terms: -1.118033989 + -2.5 = -3.618033989 t = -3.618033989 Simplifying t = -3.618033989Solution
The solution to the problem is based on the solutions from the subproblems. t = {-1.381966011, -3.618033989}
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