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Simplifying 10 = 32t + -5t2 Solving 10 = 32t + -5t2 Solving for variable 't'. Reorder the terms: 10 + -32t + 5t2 = 32t + -32t + -5t2 + 5t2 Combine like terms: 32t + -32t = 0 10 + -32t + 5t2 = 0 + -5t2 + 5t2 10 + -32t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 10 + -32t + 5t2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 2 + -6.4t + t2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + -6.4t + -2 + t2 = 0 + -2 Reorder the terms: 2 + -2 + -6.4t + t2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -6.4t + t2 = 0 + -2 -6.4t + t2 = 0 + -2 Combine like terms: 0 + -2 = -2 -6.4t + t2 = -2 The t term is -6.4t. Take half its coefficient (-3.2). Square it (10.24) and add it to both sides. Add '10.24' to each side of the equation. -6.4t + 10.24 + t2 = -2 + 10.24 Reorder the terms: 10.24 + -6.4t + t2 = -2 + 10.24 Combine like terms: -2 + 10.24 = 8.24 10.24 + -6.4t + t2 = 8.24 Factor a perfect square on the left side: (t + -3.2)(t + -3.2) = 8.24 Calculate the square root of the right side: 2.870540019 Break this problem into two subproblems by setting (t + -3.2) equal to 2.870540019 and -2.870540019.Subproblem 1
t + -3.2 = 2.870540019 Simplifying t + -3.2 = 2.870540019 Reorder the terms: -3.2 + t = 2.870540019 Solving -3.2 + t = 2.870540019 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.2' to each side of the equation. -3.2 + 3.2 + t = 2.870540019 + 3.2 Combine like terms: -3.2 + 3.2 = 0.0 0.0 + t = 2.870540019 + 3.2 t = 2.870540019 + 3.2 Combine like terms: 2.870540019 + 3.2 = 6.070540019 t = 6.070540019 Simplifying t = 6.070540019Subproblem 2
t + -3.2 = -2.870540019 Simplifying t + -3.2 = -2.870540019 Reorder the terms: -3.2 + t = -2.870540019 Solving -3.2 + t = -2.870540019 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.2' to each side of the equation. -3.2 + 3.2 + t = -2.870540019 + 3.2 Combine like terms: -3.2 + 3.2 = 0.0 0.0 + t = -2.870540019 + 3.2 t = -2.870540019 + 3.2 Combine like terms: -2.870540019 + 3.2 = 0.329459981 t = 0.329459981 Simplifying t = 0.329459981Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.070540019, 0.329459981}
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