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Simplifying 10 = 20t + -5t2 Solving 10 = 20t + -5t2 Solving for variable 't'. Reorder the terms: 10 + -20t + 5t2 = 20t + -20t + -5t2 + 5t2 Combine like terms: 20t + -20t = 0 10 + -20t + 5t2 = 0 + -5t2 + 5t2 10 + -20t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 10 + -20t + 5t2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(2 + -4t + t2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(2 + -4t + t2)' equal to zero and attempt to solve: Simplifying 2 + -4t + t2 = 0 Solving 2 + -4t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-2' to each side of the equation. 2 + -4t + -2 + t2 = 0 + -2 Reorder the terms: 2 + -2 + -4t + t2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -4t + t2 = 0 + -2 -4t + t2 = 0 + -2 Combine like terms: 0 + -2 = -2 -4t + t2 = -2 The t term is -4t. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4t + 4 + t2 = -2 + 4 Reorder the terms: 4 + -4t + t2 = -2 + 4 Combine like terms: -2 + 4 = 2 4 + -4t + t2 = 2 Factor a perfect square on the left side: (t + -2)(t + -2) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (t + -2) equal to 1.414213562 and -1.414213562.Subproblem 1
t + -2 = 1.414213562 Simplifying t + -2 = 1.414213562 Reorder the terms: -2 + t = 1.414213562 Solving -2 + t = 1.414213562 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = 1.414213562 + 2 Combine like terms: -2 + 2 = 0 0 + t = 1.414213562 + 2 t = 1.414213562 + 2 Combine like terms: 1.414213562 + 2 = 3.414213562 t = 3.414213562 Simplifying t = 3.414213562Subproblem 2
t + -2 = -1.414213562 Simplifying t + -2 = -1.414213562 Reorder the terms: -2 + t = -1.414213562 Solving -2 + t = -1.414213562 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = -1.414213562 + 2 Combine like terms: -2 + 2 = 0 0 + t = -1.414213562 + 2 t = -1.414213562 + 2 Combine like terms: -1.414213562 + 2 = 0.585786438 t = 0.585786438 Simplifying t = 0.585786438Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.414213562, 0.585786438}Solution
t = {3.414213562, 0.585786438}
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