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Simplifying -1t2 + [10t + 2(2 + -4t2)] = 0 -1t2 + [10t + (2 * 2 + -4t2 * 2)] = 0 -1t2 + [10t + (4 + -8t2)] = 0 Reorder the terms: -1t2 + [4 + 10t + -8t2] = 0 Remove brackets around [4 + 10t + -8t2] -1t2 + 4 + 10t + -8t2 = 0 Reorder the terms: 4 + 10t + -1t2 + -8t2 = 0 Combine like terms: -1t2 + -8t2 = -9t2 4 + 10t + -9t2 = 0 Solving 4 + 10t + -9t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -0.4444444444 + -1.111111111t + t2 = 0 Move the constant term to the right: Add '0.4444444444' to each side of the equation. -0.4444444444 + -1.111111111t + 0.4444444444 + t2 = 0 + 0.4444444444 Reorder the terms: -0.4444444444 + 0.4444444444 + -1.111111111t + t2 = 0 + 0.4444444444 Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000 0.0000000000 + -1.111111111t + t2 = 0 + 0.4444444444 -1.111111111t + t2 = 0 + 0.4444444444 Combine like terms: 0 + 0.4444444444 = 0.4444444444 -1.111111111t + t2 = 0.4444444444 The t term is -1.111111111t. Take half its coefficient (-0.5555555555). Square it (0.3086419752) and add it to both sides. Add '0.3086419752' to each side of the equation. -1.111111111t + 0.3086419752 + t2 = 0.4444444444 + 0.3086419752 Reorder the terms: 0.3086419752 + -1.111111111t + t2 = 0.4444444444 + 0.3086419752 Combine like terms: 0.4444444444 + 0.3086419752 = 0.7530864196 0.3086419752 + -1.111111111t + t2 = 0.7530864196 Factor a perfect square on the left side: (t + -0.5555555555)(t + -0.5555555555) = 0.7530864196 Calculate the square root of the right side: 0.867805519 Break this problem into two subproblems by setting (t + -0.5555555555) equal to 0.867805519 and -0.867805519.Subproblem 1
t + -0.5555555555 = 0.867805519 Simplifying t + -0.5555555555 = 0.867805519 Reorder the terms: -0.5555555555 + t = 0.867805519 Solving -0.5555555555 + t = 0.867805519 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5555555555' to each side of the equation. -0.5555555555 + 0.5555555555 + t = 0.867805519 + 0.5555555555 Combine like terms: -0.5555555555 + 0.5555555555 = 0.0000000000 0.0000000000 + t = 0.867805519 + 0.5555555555 t = 0.867805519 + 0.5555555555 Combine like terms: 0.867805519 + 0.5555555555 = 1.4233610745 t = 1.4233610745 Simplifying t = 1.4233610745Subproblem 2
t + -0.5555555555 = -0.867805519 Simplifying t + -0.5555555555 = -0.867805519 Reorder the terms: -0.5555555555 + t = -0.867805519 Solving -0.5555555555 + t = -0.867805519 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5555555555' to each side of the equation. -0.5555555555 + 0.5555555555 + t = -0.867805519 + 0.5555555555 Combine like terms: -0.5555555555 + 0.5555555555 = 0.0000000000 0.0000000000 + t = -0.867805519 + 0.5555555555 t = -0.867805519 + 0.5555555555 Combine like terms: -0.867805519 + 0.5555555555 = -0.3122499635 t = -0.3122499635 Simplifying t = -0.3122499635Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.4233610745, -0.3122499635}
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