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Simplifying 16t2 + 80t + 3 = 0 Reorder the terms: 3 + 80t + 16t2 = 0 Solving 3 + 80t + 16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.1875 + 5t + t2 = 0 Move the constant term to the right: Add '-0.1875' to each side of the equation. 0.1875 + 5t + -0.1875 + t2 = 0 + -0.1875 Reorder the terms: 0.1875 + -0.1875 + 5t + t2 = 0 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + 5t + t2 = 0 + -0.1875 5t + t2 = 0 + -0.1875 Combine like terms: 0 + -0.1875 = -0.1875 5t + t2 = -0.1875 The t term is 5t. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5t + 6.25 + t2 = -0.1875 + 6.25 Reorder the terms: 6.25 + 5t + t2 = -0.1875 + 6.25 Combine like terms: -0.1875 + 6.25 = 6.0625 6.25 + 5t + t2 = 6.0625 Factor a perfect square on the left side: (t + 2.5)(t + 2.5) = 6.0625 Calculate the square root of the right side: 2.46221445 Break this problem into two subproblems by setting (t + 2.5) equal to 2.46221445 and -2.46221445.Subproblem 1
t + 2.5 = 2.46221445 Simplifying t + 2.5 = 2.46221445 Reorder the terms: 2.5 + t = 2.46221445 Solving 2.5 + t = 2.46221445 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + t = 2.46221445 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = 2.46221445 + -2.5 t = 2.46221445 + -2.5 Combine like terms: 2.46221445 + -2.5 = -0.03778555 t = -0.03778555 Simplifying t = -0.03778555Subproblem 2
t + 2.5 = -2.46221445 Simplifying t + 2.5 = -2.46221445 Reorder the terms: 2.5 + t = -2.46221445 Solving 2.5 + t = -2.46221445 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + t = -2.46221445 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = -2.46221445 + -2.5 t = -2.46221445 + -2.5 Combine like terms: -2.46221445 + -2.5 = -4.96221445 t = -4.96221445 Simplifying t = -4.96221445Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.03778555, -4.96221445}
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