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Simplifying 490 = 16t2 + 8t + 96 Reorder the terms: 490 = 96 + 8t + 16t2 Solving 490 = 96 + 8t + 16t2 Solving for variable 't'. Combine like terms: 490 + -96 = 394 394 + -8t + -16t2 = 96 + 8t + 16t2 + -96 + -8t + -16t2 Reorder the terms: 394 + -8t + -16t2 = 96 + -96 + 8t + -8t + 16t2 + -16t2 Combine like terms: 96 + -96 = 0 394 + -8t + -16t2 = 0 + 8t + -8t + 16t2 + -16t2 394 + -8t + -16t2 = 8t + -8t + 16t2 + -16t2 Combine like terms: 8t + -8t = 0 394 + -8t + -16t2 = 0 + 16t2 + -16t2 394 + -8t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 394 + -8t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(197 + -4t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(197 + -4t + -8t2)' equal to zero and attempt to solve: Simplifying 197 + -4t + -8t2 = 0 Solving 197 + -4t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -24.625 + 0.5t + t2 = 0 Move the constant term to the right: Add '24.625' to each side of the equation. -24.625 + 0.5t + 24.625 + t2 = 0 + 24.625 Reorder the terms: -24.625 + 24.625 + 0.5t + t2 = 0 + 24.625 Combine like terms: -24.625 + 24.625 = 0.000 0.000 + 0.5t + t2 = 0 + 24.625 0.5t + t2 = 0 + 24.625 Combine like terms: 0 + 24.625 = 24.625 0.5t + t2 = 24.625 The t term is 0.5t. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5t + 0.0625 + t2 = 24.625 + 0.0625 Reorder the terms: 0.0625 + 0.5t + t2 = 24.625 + 0.0625 Combine like terms: 24.625 + 0.0625 = 24.6875 0.0625 + 0.5t + t2 = 24.6875 Factor a perfect square on the left side: (t + 0.25)(t + 0.25) = 24.6875 Calculate the square root of the right side: 4.968651729 Break this problem into two subproblems by setting (t + 0.25) equal to 4.968651729 and -4.968651729.Subproblem 1
t + 0.25 = 4.968651729 Simplifying t + 0.25 = 4.968651729 Reorder the terms: 0.25 + t = 4.968651729 Solving 0.25 + t = 4.968651729 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + t = 4.968651729 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + t = 4.968651729 + -0.25 t = 4.968651729 + -0.25 Combine like terms: 4.968651729 + -0.25 = 4.718651729 t = 4.718651729 Simplifying t = 4.718651729Subproblem 2
t + 0.25 = -4.968651729 Simplifying t + 0.25 = -4.968651729 Reorder the terms: 0.25 + t = -4.968651729 Solving 0.25 + t = -4.968651729 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + t = -4.968651729 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + t = -4.968651729 + -0.25 t = -4.968651729 + -0.25 Combine like terms: -4.968651729 + -0.25 = -5.218651729 t = -5.218651729 Simplifying t = -5.218651729Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.718651729, -5.218651729}Solution
t = {4.718651729, -5.218651729}
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