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Simplifying 0 = 64t2 + 64t + 5 Reorder the terms: 0 = 5 + 64t + 64t2 Solving 0 = 5 + 64t + 64t2 Solving for variable 't'. Combine like terms: 0 + -5 = -5 -5 + -64t + -64t2 = 5 + 64t + 64t2 + -5 + -64t + -64t2 Reorder the terms: -5 + -64t + -64t2 = 5 + -5 + 64t + -64t + 64t2 + -64t2 Combine like terms: 5 + -5 = 0 -5 + -64t + -64t2 = 0 + 64t + -64t + 64t2 + -64t2 -5 + -64t + -64t2 = 64t + -64t + 64t2 + -64t2 Combine like terms: 64t + -64t = 0 -5 + -64t + -64t2 = 0 + 64t2 + -64t2 -5 + -64t + -64t2 = 64t2 + -64t2 Combine like terms: 64t2 + -64t2 = 0 -5 + -64t + -64t2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(5 + 64t + 64t2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(5 + 64t + 64t2)' equal to zero and attempt to solve: Simplifying 5 + 64t + 64t2 = 0 Solving 5 + 64t + 64t2 = 0 Begin completing the square. Divide all terms by 64 the coefficient of the squared term: Divide each side by '64'. 0.078125 + t + t2 = 0 Move the constant term to the right: Add '-0.078125' to each side of the equation. 0.078125 + t + -0.078125 + t2 = 0 + -0.078125 Reorder the terms: 0.078125 + -0.078125 + t + t2 = 0 + -0.078125 Combine like terms: 0.078125 + -0.078125 = 0.000000 0.000000 + t + t2 = 0 + -0.078125 t + t2 = 0 + -0.078125 Combine like terms: 0 + -0.078125 = -0.078125 t + t2 = -0.078125 The t term is t. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + t2 = -0.078125 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + t2 = -0.078125 + 0.25 Combine like terms: -0.078125 + 0.25 = 0.171875 1.25 + t2 = 0.171875 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 0.171875 Calculate the square root of the right side: 0.414578099 Break this problem into two subproblems by setting (t + 0.5) equal to 0.414578099 and -0.414578099.Subproblem 1
t + 0.5 = 0.414578099 Simplifying t + 0.5 = 0.414578099 Reorder the terms: 0.5 + t = 0.414578099 Solving 0.5 + t = 0.414578099 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = 0.414578099 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 0.414578099 + -0.5 t = 0.414578099 + -0.5 Combine like terms: 0.414578099 + -0.5 = -0.085421901 t = -0.085421901 Simplifying t = -0.085421901Subproblem 2
t + 0.5 = -0.414578099 Simplifying t + 0.5 = -0.414578099 Reorder the terms: 0.5 + t = -0.414578099 Solving 0.5 + t = -0.414578099 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = -0.414578099 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -0.414578099 + -0.5 t = -0.414578099 + -0.5 Combine like terms: -0.414578099 + -0.5 = -0.914578099 t = -0.914578099 Simplifying t = -0.914578099Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.085421901, -0.914578099}Solution
t = {-0.085421901, -0.914578099}
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