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Simplifying 10 = 5t2 + 2t Reorder the terms: 10 = 2t + 5t2 Solving 10 = 2t + 5t2 Solving for variable 't'. Reorder the terms: 10 + -2t + -5t2 = 2t + -2t + 5t2 + -5t2 Combine like terms: 2t + -2t = 0 10 + -2t + -5t2 = 0 + 5t2 + -5t2 10 + -2t + -5t2 = 5t2 + -5t2 Combine like terms: 5t2 + -5t2 = 0 10 + -2t + -5t2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -2 + 0.4t + t2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 0.4t + 2 + t2 = 0 + 2 Reorder the terms: -2 + 2 + 0.4t + t2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 0.4t + t2 = 0 + 2 0.4t + t2 = 0 + 2 Combine like terms: 0 + 2 = 2 0.4t + t2 = 2 The t term is 0.4t. Take half its coefficient (0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. 0.4t + 0.04 + t2 = 2 + 0.04 Reorder the terms: 0.04 + 0.4t + t2 = 2 + 0.04 Combine like terms: 2 + 0.04 = 2.04 0.04 + 0.4t + t2 = 2.04 Factor a perfect square on the left side: (t + 0.2)(t + 0.2) = 2.04 Calculate the square root of the right side: 1.428285686 Break this problem into two subproblems by setting (t + 0.2) equal to 1.428285686 and -1.428285686.Subproblem 1
t + 0.2 = 1.428285686 Simplifying t + 0.2 = 1.428285686 Reorder the terms: 0.2 + t = 1.428285686 Solving 0.2 + t = 1.428285686 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + t = 1.428285686 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + t = 1.428285686 + -0.2 t = 1.428285686 + -0.2 Combine like terms: 1.428285686 + -0.2 = 1.228285686 t = 1.228285686 Simplifying t = 1.228285686Subproblem 2
t + 0.2 = -1.428285686 Simplifying t + 0.2 = -1.428285686 Reorder the terms: 0.2 + t = -1.428285686 Solving 0.2 + t = -1.428285686 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + t = -1.428285686 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + t = -1.428285686 + -0.2 t = -1.428285686 + -0.2 Combine like terms: -1.428285686 + -0.2 = -1.628285686 t = -1.628285686 Simplifying t = -1.628285686Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.228285686, -1.628285686}
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