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Simplifying 45 = -5t2 + 30t + 10 Reorder the terms: 45 = 10 + 30t + -5t2 Solving 45 = 10 + 30t + -5t2 Solving for variable 't'. Combine like terms: 45 + -10 = 35 35 + -30t + 5t2 = 10 + 30t + -5t2 + -10 + -30t + 5t2 Reorder the terms: 35 + -30t + 5t2 = 10 + -10 + 30t + -30t + -5t2 + 5t2 Combine like terms: 10 + -10 = 0 35 + -30t + 5t2 = 0 + 30t + -30t + -5t2 + 5t2 35 + -30t + 5t2 = 30t + -30t + -5t2 + 5t2 Combine like terms: 30t + -30t = 0 35 + -30t + 5t2 = 0 + -5t2 + 5t2 35 + -30t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 35 + -30t + 5t2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(7 + -6t + t2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(7 + -6t + t2)' equal to zero and attempt to solve: Simplifying 7 + -6t + t2 = 0 Solving 7 + -6t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-7' to each side of the equation. 7 + -6t + -7 + t2 = 0 + -7 Reorder the terms: 7 + -7 + -6t + t2 = 0 + -7 Combine like terms: 7 + -7 = 0 0 + -6t + t2 = 0 + -7 -6t + t2 = 0 + -7 Combine like terms: 0 + -7 = -7 -6t + t2 = -7 The t term is -6t. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6t + 9 + t2 = -7 + 9 Reorder the terms: 9 + -6t + t2 = -7 + 9 Combine like terms: -7 + 9 = 2 9 + -6t + t2 = 2 Factor a perfect square on the left side: (t + -3)(t + -3) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (t + -3) equal to 1.414213562 and -1.414213562.Subproblem 1
t + -3 = 1.414213562 Simplifying t + -3 = 1.414213562 Reorder the terms: -3 + t = 1.414213562 Solving -3 + t = 1.414213562 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = 1.414213562 + 3 Combine like terms: -3 + 3 = 0 0 + t = 1.414213562 + 3 t = 1.414213562 + 3 Combine like terms: 1.414213562 + 3 = 4.414213562 t = 4.414213562 Simplifying t = 4.414213562Subproblem 2
t + -3 = -1.414213562 Simplifying t + -3 = -1.414213562 Reorder the terms: -3 + t = -1.414213562 Solving -3 + t = -1.414213562 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = -1.414213562 + 3 Combine like terms: -3 + 3 = 0 0 + t = -1.414213562 + 3 t = -1.414213562 + 3 Combine like terms: -1.414213562 + 3 = 1.585786438 t = 1.585786438 Simplifying t = 1.585786438Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.414213562, 1.585786438}Solution
t = {4.414213562, 1.585786438}
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