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Simplifying 100(t) = -16t2 + 80t + 4 Reorder the terms: 100t = 4 + 80t + -16t2 Solving 100t = 4 + 80t + -16t2 Solving for variable 't'. Reorder the terms: -4 + 100t + -80t + 16t2 = 4 + 80t + -16t2 + -4 + -80t + 16t2 Combine like terms: 100t + -80t = 20t -4 + 20t + 16t2 = 4 + 80t + -16t2 + -4 + -80t + 16t2 Reorder the terms: -4 + 20t + 16t2 = 4 + -4 + 80t + -80t + -16t2 + 16t2 Combine like terms: 4 + -4 = 0 -4 + 20t + 16t2 = 0 + 80t + -80t + -16t2 + 16t2 -4 + 20t + 16t2 = 80t + -80t + -16t2 + 16t2 Combine like terms: 80t + -80t = 0 -4 + 20t + 16t2 = 0 + -16t2 + 16t2 -4 + 20t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -4 + 20t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-1 + 5t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-1 + 5t + 4t2)' equal to zero and attempt to solve: Simplifying -1 + 5t + 4t2 = 0 Solving -1 + 5t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + 1.25t + t2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + 1.25t + 0.25 + t2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + 1.25t + t2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + 1.25t + t2 = 0 + 0.25 1.25t + t2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 1.25t + t2 = 0.25 The t term is 1.25t. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25t + 0.390625 + t2 = 0.25 + 0.390625 Reorder the terms: 0.390625 + 1.25t + t2 = 0.25 + 0.390625 Combine like terms: 0.25 + 0.390625 = 0.640625 0.390625 + 1.25t + t2 = 0.640625 Factor a perfect square on the left side: (t + 0.625)(t + 0.625) = 0.640625 Calculate the square root of the right side: 0.80039053 Break this problem into two subproblems by setting (t + 0.625) equal to 0.80039053 and -0.80039053.Subproblem 1
t + 0.625 = 0.80039053 Simplifying t + 0.625 = 0.80039053 Reorder the terms: 0.625 + t = 0.80039053 Solving 0.625 + t = 0.80039053 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + t = 0.80039053 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + t = 0.80039053 + -0.625 t = 0.80039053 + -0.625 Combine like terms: 0.80039053 + -0.625 = 0.17539053 t = 0.17539053 Simplifying t = 0.17539053Subproblem 2
t + 0.625 = -0.80039053 Simplifying t + 0.625 = -0.80039053 Reorder the terms: 0.625 + t = -0.80039053 Solving 0.625 + t = -0.80039053 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + t = -0.80039053 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + t = -0.80039053 + -0.625 t = -0.80039053 + -0.625 Combine like terms: -0.80039053 + -0.625 = -1.42539053 t = -1.42539053 Simplifying t = -1.42539053Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.17539053, -1.42539053}Solution
t = {0.17539053, -1.42539053}
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