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Simplifying 2.5t2 + 80t = 1000 Reorder the terms: 80t + 2.5t2 = 1000 Solving 80t + 2.5t2 = 1000 Solving for variable 't'. Reorder the terms: -1000 + 80t + 2.5t2 = 1000 + -1000 Combine like terms: 1000 + -1000 = 0 -1000 + 80t + 2.5t2 = 0 Begin completing the square. Divide all terms by 2.5 the coefficient of the squared term: Divide each side by '2.5'. -400 + 32t + t2 = 0 Move the constant term to the right: Add '400' to each side of the equation. -400 + 32t + 400 + t2 = 0 + 400 Reorder the terms: -400 + 400 + 32t + t2 = 0 + 400 Combine like terms: -400 + 400 = 0 0 + 32t + t2 = 0 + 400 32t + t2 = 0 + 400 Combine like terms: 0 + 400 = 400 32t + t2 = 400 The t term is 32t. Take half its coefficient (16). Square it (256) and add it to both sides. Add '256' to each side of the equation. 32t + 256 + t2 = 400 + 256 Reorder the terms: 256 + 32t + t2 = 400 + 256 Combine like terms: 400 + 256 = 656 256 + 32t + t2 = 656 Factor a perfect square on the left side: (t + 16)(t + 16) = 656 Calculate the square root of the right side: 25.61249695 Break this problem into two subproblems by setting (t + 16) equal to 25.61249695 and -25.61249695.Subproblem 1
t + 16 = 25.61249695 Simplifying t + 16 = 25.61249695 Reorder the terms: 16 + t = 25.61249695 Solving 16 + t = 25.61249695 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-16' to each side of the equation. 16 + -16 + t = 25.61249695 + -16 Combine like terms: 16 + -16 = 0 0 + t = 25.61249695 + -16 t = 25.61249695 + -16 Combine like terms: 25.61249695 + -16 = 9.61249695 t = 9.61249695 Simplifying t = 9.61249695Subproblem 2
t + 16 = -25.61249695 Simplifying t + 16 = -25.61249695 Reorder the terms: 16 + t = -25.61249695 Solving 16 + t = -25.61249695 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-16' to each side of the equation. 16 + -16 + t = -25.61249695 + -16 Combine like terms: 16 + -16 = 0 0 + t = -25.61249695 + -16 t = -25.61249695 + -16 Combine like terms: -25.61249695 + -16 = -41.61249695 t = -41.61249695 Simplifying t = -41.61249695Solution
The solution to the problem is based on the solutions from the subproblems. t = {9.61249695, -41.61249695}
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