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Simplifying (-3t2 + 5t + -6) + (7t2 + 4t + 1) = 0 Reorder the terms: (-6 + 5t + -3t2) + (7t2 + 4t + 1) = 0 Remove parenthesis around (-6 + 5t + -3t2) -6 + 5t + -3t2 + (7t2 + 4t + 1) = 0 Reorder the terms: -6 + 5t + -3t2 + (1 + 4t + 7t2) = 0 Remove parenthesis around (1 + 4t + 7t2) -6 + 5t + -3t2 + 1 + 4t + 7t2 = 0 Reorder the terms: -6 + 1 + 5t + 4t + -3t2 + 7t2 = 0 Combine like terms: -6 + 1 = -5 -5 + 5t + 4t + -3t2 + 7t2 = 0 Combine like terms: 5t + 4t = 9t -5 + 9t + -3t2 + 7t2 = 0 Combine like terms: -3t2 + 7t2 = 4t2 -5 + 9t + 4t2 = 0 Solving -5 + 9t + 4t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.25 + 2.25t + t2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + 2.25t + 1.25 + t2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + 2.25t + t2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + 2.25t + t2 = 0 + 1.25 2.25t + t2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 2.25t + t2 = 1.25 The t term is 2.25t. Take half its coefficient (1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. 2.25t + 1.265625 + t2 = 1.25 + 1.265625 Reorder the terms: 1.265625 + 2.25t + t2 = 1.25 + 1.265625 Combine like terms: 1.25 + 1.265625 = 2.515625 1.265625 + 2.25t + t2 = 2.515625 Factor a perfect square on the left side: (t + 1.125)(t + 1.125) = 2.515625 Calculate the square root of the right side: 1.586072193 Break this problem into two subproblems by setting (t + 1.125) equal to 1.586072193 and -1.586072193.Subproblem 1
t + 1.125 = 1.586072193 Simplifying t + 1.125 = 1.586072193 Reorder the terms: 1.125 + t = 1.586072193 Solving 1.125 + t = 1.586072193 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + t = 1.586072193 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + t = 1.586072193 + -1.125 t = 1.586072193 + -1.125 Combine like terms: 1.586072193 + -1.125 = 0.461072193 t = 0.461072193 Simplifying t = 0.461072193Subproblem 2
t + 1.125 = -1.586072193 Simplifying t + 1.125 = -1.586072193 Reorder the terms: 1.125 + t = -1.586072193 Solving 1.125 + t = -1.586072193 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + t = -1.586072193 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + t = -1.586072193 + -1.125 t = -1.586072193 + -1.125 Combine like terms: -1.586072193 + -1.125 = -2.711072193 t = -2.711072193 Simplifying t = -2.711072193Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.461072193, -2.711072193}
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