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Simplifying 25 = -16t2 + 5t + 92 Reorder the terms: 25 = 92 + 5t + -16t2 Solving 25 = 92 + 5t + -16t2 Solving for variable 't'. Combine like terms: 25 + -92 = -67 -67 + -5t + 16t2 = 92 + 5t + -16t2 + -92 + -5t + 16t2 Reorder the terms: -67 + -5t + 16t2 = 92 + -92 + 5t + -5t + -16t2 + 16t2 Combine like terms: 92 + -92 = 0 -67 + -5t + 16t2 = 0 + 5t + -5t + -16t2 + 16t2 -67 + -5t + 16t2 = 5t + -5t + -16t2 + 16t2 Combine like terms: 5t + -5t = 0 -67 + -5t + 16t2 = 0 + -16t2 + 16t2 -67 + -5t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -67 + -5t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -4.1875 + -0.3125t + t2 = 0 Move the constant term to the right: Add '4.1875' to each side of the equation. -4.1875 + -0.3125t + 4.1875 + t2 = 0 + 4.1875 Reorder the terms: -4.1875 + 4.1875 + -0.3125t + t2 = 0 + 4.1875 Combine like terms: -4.1875 + 4.1875 = 0.0000 0.0000 + -0.3125t + t2 = 0 + 4.1875 -0.3125t + t2 = 0 + 4.1875 Combine like terms: 0 + 4.1875 = 4.1875 -0.3125t + t2 = 4.1875 The t term is -0.3125t. Take half its coefficient (-0.15625). Square it (0.0244140625) and add it to both sides. Add '0.0244140625' to each side of the equation. -0.3125t + 0.0244140625 + t2 = 4.1875 + 0.0244140625 Reorder the terms: 0.0244140625 + -0.3125t + t2 = 4.1875 + 0.0244140625 Combine like terms: 4.1875 + 0.0244140625 = 4.2119140625 0.0244140625 + -0.3125t + t2 = 4.2119140625 Factor a perfect square on the left side: (t + -0.15625)(t + -0.15625) = 4.2119140625 Calculate the square root of the right side: 2.052294828 Break this problem into two subproblems by setting (t + -0.15625) equal to 2.052294828 and -2.052294828.Subproblem 1
t + -0.15625 = 2.052294828 Simplifying t + -0.15625 = 2.052294828 Reorder the terms: -0.15625 + t = 2.052294828 Solving -0.15625 + t = 2.052294828 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.15625' to each side of the equation. -0.15625 + 0.15625 + t = 2.052294828 + 0.15625 Combine like terms: -0.15625 + 0.15625 = 0.00000 0.00000 + t = 2.052294828 + 0.15625 t = 2.052294828 + 0.15625 Combine like terms: 2.052294828 + 0.15625 = 2.208544828 t = 2.208544828 Simplifying t = 2.208544828Subproblem 2
t + -0.15625 = -2.052294828 Simplifying t + -0.15625 = -2.052294828 Reorder the terms: -0.15625 + t = -2.052294828 Solving -0.15625 + t = -2.052294828 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.15625' to each side of the equation. -0.15625 + 0.15625 + t = -2.052294828 + 0.15625 Combine like terms: -0.15625 + 0.15625 = 0.00000 0.00000 + t = -2.052294828 + 0.15625 t = -2.052294828 + 0.15625 Combine like terms: -2.052294828 + 0.15625 = -1.896044828 t = -1.896044828 Simplifying t = -1.896044828Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.208544828, -1.896044828}
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