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Simplifying 16t2 + 6t + -500 = 0 Reorder the terms: -500 + 6t + 16t2 = 0 Solving -500 + 6t + 16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(-250 + 3t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-250 + 3t + 8t2)' equal to zero and attempt to solve: Simplifying -250 + 3t + 8t2 = 0 Solving -250 + 3t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -31.25 + 0.375t + t2 = 0 Move the constant term to the right: Add '31.25' to each side of the equation. -31.25 + 0.375t + 31.25 + t2 = 0 + 31.25 Reorder the terms: -31.25 + 31.25 + 0.375t + t2 = 0 + 31.25 Combine like terms: -31.25 + 31.25 = 0.00 0.00 + 0.375t + t2 = 0 + 31.25 0.375t + t2 = 0 + 31.25 Combine like terms: 0 + 31.25 = 31.25 0.375t + t2 = 31.25 The t term is 0.375t. Take half its coefficient (0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. 0.375t + 0.03515625 + t2 = 31.25 + 0.03515625 Reorder the terms: 0.03515625 + 0.375t + t2 = 31.25 + 0.03515625 Combine like terms: 31.25 + 0.03515625 = 31.28515625 0.03515625 + 0.375t + t2 = 31.28515625 Factor a perfect square on the left side: (t + 0.1875)(t + 0.1875) = 31.28515625 Calculate the square root of the right side: 5.59331353 Break this problem into two subproblems by setting (t + 0.1875) equal to 5.59331353 and -5.59331353.Subproblem 1
t + 0.1875 = 5.59331353 Simplifying t + 0.1875 = 5.59331353 Reorder the terms: 0.1875 + t = 5.59331353 Solving 0.1875 + t = 5.59331353 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + t = 5.59331353 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + t = 5.59331353 + -0.1875 t = 5.59331353 + -0.1875 Combine like terms: 5.59331353 + -0.1875 = 5.40581353 t = 5.40581353 Simplifying t = 5.40581353Subproblem 2
t + 0.1875 = -5.59331353 Simplifying t + 0.1875 = -5.59331353 Reorder the terms: 0.1875 + t = -5.59331353 Solving 0.1875 + t = -5.59331353 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + t = -5.59331353 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + t = -5.59331353 + -0.1875 t = -5.59331353 + -0.1875 Combine like terms: -5.59331353 + -0.1875 = -5.78081353 t = -5.78081353 Simplifying t = -5.78081353Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.40581353, -5.78081353}Solution
t = {5.40581353, -5.78081353}
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