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Simplifying 0 = 16t2 + 125t + 5 Reorder the terms: 0 = 5 + 125t + 16t2 Solving 0 = 5 + 125t + 16t2 Solving for variable 't'. Combine like terms: 0 + -5 = -5 -5 + -125t + -16t2 = 5 + 125t + 16t2 + -5 + -125t + -16t2 Reorder the terms: -5 + -125t + -16t2 = 5 + -5 + 125t + -125t + 16t2 + -16t2 Combine like terms: 5 + -5 = 0 -5 + -125t + -16t2 = 0 + 125t + -125t + 16t2 + -16t2 -5 + -125t + -16t2 = 125t + -125t + 16t2 + -16t2 Combine like terms: 125t + -125t = 0 -5 + -125t + -16t2 = 0 + 16t2 + -16t2 -5 + -125t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -5 + -125t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(5 + 125t + 16t2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(5 + 125t + 16t2)' equal to zero and attempt to solve: Simplifying 5 + 125t + 16t2 = 0 Solving 5 + 125t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.3125 + 7.8125t + t2 = 0 Move the constant term to the right: Add '-0.3125' to each side of the equation. 0.3125 + 7.8125t + -0.3125 + t2 = 0 + -0.3125 Reorder the terms: 0.3125 + -0.3125 + 7.8125t + t2 = 0 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + 7.8125t + t2 = 0 + -0.3125 7.8125t + t2 = 0 + -0.3125 Combine like terms: 0 + -0.3125 = -0.3125 7.8125t + t2 = -0.3125 The t term is 7.8125t. Take half its coefficient (3.90625). Square it (15.25878906) and add it to both sides. Add '15.25878906' to each side of the equation. 7.8125t + 15.25878906 + t2 = -0.3125 + 15.25878906 Reorder the terms: 15.25878906 + 7.8125t + t2 = -0.3125 + 15.25878906 Combine like terms: -0.3125 + 15.25878906 = 14.94628906 15.25878906 + 7.8125t + t2 = 14.94628906 Factor a perfect square on the left side: (t + 3.90625)(t + 3.90625) = 14.94628906 Calculate the square root of the right side: 3.866043075 Break this problem into two subproblems by setting (t + 3.90625) equal to 3.866043075 and -3.866043075.Subproblem 1
t + 3.90625 = 3.866043075 Simplifying t + 3.90625 = 3.866043075 Reorder the terms: 3.90625 + t = 3.866043075 Solving 3.90625 + t = 3.866043075 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.90625' to each side of the equation. 3.90625 + -3.90625 + t = 3.866043075 + -3.90625 Combine like terms: 3.90625 + -3.90625 = 0.00000 0.00000 + t = 3.866043075 + -3.90625 t = 3.866043075 + -3.90625 Combine like terms: 3.866043075 + -3.90625 = -0.040206925 t = -0.040206925 Simplifying t = -0.040206925Subproblem 2
t + 3.90625 = -3.866043075 Simplifying t + 3.90625 = -3.866043075 Reorder the terms: 3.90625 + t = -3.866043075 Solving 3.90625 + t = -3.866043075 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.90625' to each side of the equation. 3.90625 + -3.90625 + t = -3.866043075 + -3.90625 Combine like terms: 3.90625 + -3.90625 = 0.00000 0.00000 + t = -3.866043075 + -3.90625 t = -3.866043075 + -3.90625 Combine like terms: -3.866043075 + -3.90625 = -7.772293075 t = -7.772293075 Simplifying t = -7.772293075Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.040206925, -7.772293075}Solution
t = {-0.040206925, -7.772293075}
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