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Simplifying -2t2 + -20t + -46 = 0 Reorder the terms: -46 + -20t + -2t2 = 0 Solving -46 + -20t + -2t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '-2'. -2(23 + 10t + t2) = 0 Ignore the factor -2.Subproblem 1
Set the factor '(23 + 10t + t2)' equal to zero and attempt to solve: Simplifying 23 + 10t + t2 = 0 Solving 23 + 10t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-23' to each side of the equation. 23 + 10t + -23 + t2 = 0 + -23 Reorder the terms: 23 + -23 + 10t + t2 = 0 + -23 Combine like terms: 23 + -23 = 0 0 + 10t + t2 = 0 + -23 10t + t2 = 0 + -23 Combine like terms: 0 + -23 = -23 10t + t2 = -23 The t term is 10t. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10t + 25 + t2 = -23 + 25 Reorder the terms: 25 + 10t + t2 = -23 + 25 Combine like terms: -23 + 25 = 2 25 + 10t + t2 = 2 Factor a perfect square on the left side: (t + 5)(t + 5) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (t + 5) equal to 1.414213562 and -1.414213562.Subproblem 1
t + 5 = 1.414213562 Simplifying t + 5 = 1.414213562 Reorder the terms: 5 + t = 1.414213562 Solving 5 + t = 1.414213562 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + t = 1.414213562 + -5 Combine like terms: 5 + -5 = 0 0 + t = 1.414213562 + -5 t = 1.414213562 + -5 Combine like terms: 1.414213562 + -5 = -3.585786438 t = -3.585786438 Simplifying t = -3.585786438Subproblem 2
t + 5 = -1.414213562 Simplifying t + 5 = -1.414213562 Reorder the terms: 5 + t = -1.414213562 Solving 5 + t = -1.414213562 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + t = -1.414213562 + -5 Combine like terms: 5 + -5 = 0 0 + t = -1.414213562 + -5 t = -1.414213562 + -5 Combine like terms: -1.414213562 + -5 = -6.414213562 t = -6.414213562 Simplifying t = -6.414213562Solution
The solution to the problem is based on the solutions from the subproblems. t = {-3.585786438, -6.414213562}Solution
t = {-3.585786438, -6.414213562}
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