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Simplifying 0 = 80t + -16t2 + 3.5 Reorder the terms: 0 = 3.5 + 80t + -16t2 Solving 0 = 3.5 + 80t + -16t2 Solving for variable 't'. Combine like terms: 0 + -3.5 = -3.5 -3.5 + -80t + 16t2 = 3.5 + 80t + -16t2 + -3.5 + -80t + 16t2 Reorder the terms: -3.5 + -80t + 16t2 = 3.5 + -3.5 + 80t + -80t + -16t2 + 16t2 Combine like terms: 3.5 + -3.5 = 0.0 -3.5 + -80t + 16t2 = 0.0 + 80t + -80t + -16t2 + 16t2 -3.5 + -80t + 16t2 = 80t + -80t + -16t2 + 16t2 Combine like terms: 80t + -80t = 0 -3.5 + -80t + 16t2 = 0 + -16t2 + 16t2 -3.5 + -80t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -3.5 + -80t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.21875 + -5t + t2 = 0 Move the constant term to the right: Add '0.21875' to each side of the equation. -0.21875 + -5t + 0.21875 + t2 = 0 + 0.21875 Reorder the terms: -0.21875 + 0.21875 + -5t + t2 = 0 + 0.21875 Combine like terms: -0.21875 + 0.21875 = 0.00000 0.00000 + -5t + t2 = 0 + 0.21875 -5t + t2 = 0 + 0.21875 Combine like terms: 0 + 0.21875 = 0.21875 -5t + t2 = 0.21875 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 = 0.21875 + 6.25 Reorder the terms: 6.25 + -5t + t2 = 0.21875 + 6.25 Combine like terms: 0.21875 + 6.25 = 6.46875 6.25 + -5t + t2 = 6.46875 Factor a perfect square on the left side: (t + -2.5)(t + -2.5) = 6.46875 Calculate the square root of the right side: 2.543373744 Break this problem into two subproblems by setting (t + -2.5) equal to 2.543373744 and -2.543373744.Subproblem 1
t + -2.5 = 2.543373744 Simplifying t + -2.5 = 2.543373744 Reorder the terms: -2.5 + t = 2.543373744 Solving -2.5 + t = 2.543373744 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 = 2.543373744 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = 2.543373744 + 2.5 t = 2.543373744 + 2.5 Combine like terms: 2.543373744 + 2.5 = 5.043373744 t = 5.043373744 Simplifying t = 5.043373744Subproblem 2
t + -2.5 = -2.543373744 Simplifying t + -2.5 = -2.543373744 Reorder the terms: -2.5 + t = -2.543373744 Solving -2.5 + t = -2.543373744 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 = -2.543373744 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = -2.543373744 + 2.5 t = -2.543373744 + 2.5 Combine like terms: -2.543373744 + 2.5 = -0.043373744 t = -0.043373744 Simplifying t = -0.043373744Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.043373744, -0.043373744}
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