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Simplifying 0 = 1000 + -10t + -16t2 Solving 0 = 1000 + -10t + -16t2 Solving for variable 't'. Combine like terms: 0 + -1000 = -1000 -1000 + 10t + 16t2 = 1000 + -10t + -16t2 + -1000 + 10t + 16t2 Reorder the terms: -1000 + 10t + 16t2 = 1000 + -1000 + -10t + 10t + -16t2 + 16t2 Combine like terms: 1000 + -1000 = 0 -1000 + 10t + 16t2 = 0 + -10t + 10t + -16t2 + 16t2 -1000 + 10t + 16t2 = -10t + 10t + -16t2 + 16t2 Combine like terms: -10t + 10t = 0 -1000 + 10t + 16t2 = 0 + -16t2 + 16t2 -1000 + 10t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -1000 + 10t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-500 + 5t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-500 + 5t + 8t2)' equal to zero and attempt to solve: Simplifying -500 + 5t + 8t2 = 0 Solving -500 + 5t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -62.5 + 0.625t + t2 = 0 Move the constant term to the right: Add '62.5' to each side of the equation. -62.5 + 0.625t + 62.5 + t2 = 0 + 62.5 Reorder the terms: -62.5 + 62.5 + 0.625t + t2 = 0 + 62.5 Combine like terms: -62.5 + 62.5 = 0.0 0.0 + 0.625t + t2 = 0 + 62.5 0.625t + t2 = 0 + 62.5 Combine like terms: 0 + 62.5 = 62.5 0.625t + t2 = 62.5 The t term is 0.625t. Take half its coefficient (0.3125). Square it (0.09765625) and add it to both sides. Add '0.09765625' to each side of the equation. 0.625t + 0.09765625 + t2 = 62.5 + 0.09765625 Reorder the terms: 0.09765625 + 0.625t + t2 = 62.5 + 0.09765625 Combine like terms: 62.5 + 0.09765625 = 62.59765625 0.09765625 + 0.625t + t2 = 62.59765625 Factor a perfect square on the left side: (t + 0.3125)(t + 0.3125) = 62.59765625 Calculate the square root of the right side: 7.911868063 Break this problem into two subproblems by setting (t + 0.3125) equal to 7.911868063 and -7.911868063.Subproblem 1
t + 0.3125 = 7.911868063 Simplifying t + 0.3125 = 7.911868063 Reorder the terms: 0.3125 + t = 7.911868063 Solving 0.3125 + t = 7.911868063 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.3125' to each side of the equation. 0.3125 + -0.3125 + t = 7.911868063 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + t = 7.911868063 + -0.3125 t = 7.911868063 + -0.3125 Combine like terms: 7.911868063 + -0.3125 = 7.599368063 t = 7.599368063 Simplifying t = 7.599368063Subproblem 2
t + 0.3125 = -7.911868063 Simplifying t + 0.3125 = -7.911868063 Reorder the terms: 0.3125 + t = -7.911868063 Solving 0.3125 + t = -7.911868063 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.3125' to each side of the equation. 0.3125 + -0.3125 + t = -7.911868063 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + t = -7.911868063 + -0.3125 t = -7.911868063 + -0.3125 Combine like terms: -7.911868063 + -0.3125 = -8.224368063 t = -8.224368063 Simplifying t = -8.224368063Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.599368063, -8.224368063}Solution
t = {7.599368063, -8.224368063}
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