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Simplifying 5t2 + 14t = 32.8 Reorder the terms: 14t + 5t2 = 32.8 Solving 14t + 5t2 = 32.8 Solving for variable 't'. Reorder the terms: -32.8 + 14t + 5t2 = 32.8 + -32.8 Combine like terms: 32.8 + -32.8 = 0.0 -32.8 + 14t + 5t2 = 0.0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -6.56 + 2.8t + t2 = 0.0 Move the constant term to the right: Add '6.56' to each side of the equation. -6.56 + 2.8t + 6.56 + t2 = 0.0 + 6.56 Reorder the terms: -6.56 + 6.56 + 2.8t + t2 = 0.0 + 6.56 Combine like terms: -6.56 + 6.56 = 0.00 0.00 + 2.8t + t2 = 0.0 + 6.56 2.8t + t2 = 0.0 + 6.56 Combine like terms: 0.0 + 6.56 = 6.56 2.8t + t2 = 6.56 The t term is 2.8t. Take half its coefficient (1.4). Square it (1.96) and add it to both sides. Add '1.96' to each side of the equation. 2.8t + 1.96 + t2 = 6.56 + 1.96 Reorder the terms: 1.96 + 2.8t + t2 = 6.56 + 1.96 Combine like terms: 6.56 + 1.96 = 8.52 1.96 + 2.8t + t2 = 8.52 Factor a perfect square on the left side: (t + 1.4)(t + 1.4) = 8.52 Calculate the square root of the right side: 2.918903904 Break this problem into two subproblems by setting (t + 1.4) equal to 2.918903904 and -2.918903904.Subproblem 1
t + 1.4 = 2.918903904 Simplifying t + 1.4 = 2.918903904 Reorder the terms: 1.4 + t = 2.918903904 Solving 1.4 + t = 2.918903904 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.4' to each side of the equation. 1.4 + -1.4 + t = 2.918903904 + -1.4 Combine like terms: 1.4 + -1.4 = 0.0 0.0 + t = 2.918903904 + -1.4 t = 2.918903904 + -1.4 Combine like terms: 2.918903904 + -1.4 = 1.518903904 t = 1.518903904 Simplifying t = 1.518903904Subproblem 2
t + 1.4 = -2.918903904 Simplifying t + 1.4 = -2.918903904 Reorder the terms: 1.4 + t = -2.918903904 Solving 1.4 + t = -2.918903904 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.4' to each side of the equation. 1.4 + -1.4 + t = -2.918903904 + -1.4 Combine like terms: 1.4 + -1.4 = 0.0 0.0 + t = -2.918903904 + -1.4 t = -2.918903904 + -1.4 Combine like terms: -2.918903904 + -1.4 = -4.318903904 t = -4.318903904 Simplifying t = -4.318903904Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.518903904, -4.318903904}
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