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Simplifying (4t2 + -6t + -7) + (2t2 + -3t + 5) = 0 Reorder the terms: (-7 + -6t + 4t2) + (2t2 + -3t + 5) = 0 Remove parenthesis around (-7 + -6t + 4t2) -7 + -6t + 4t2 + (2t2 + -3t + 5) = 0 Reorder the terms: -7 + -6t + 4t2 + (5 + -3t + 2t2) = 0 Remove parenthesis around (5 + -3t + 2t2) -7 + -6t + 4t2 + 5 + -3t + 2t2 = 0 Reorder the terms: -7 + 5 + -6t + -3t + 4t2 + 2t2 = 0 Combine like terms: -7 + 5 = -2 -2 + -6t + -3t + 4t2 + 2t2 = 0 Combine like terms: -6t + -3t = -9t -2 + -9t + 4t2 + 2t2 = 0 Combine like terms: 4t2 + 2t2 = 6t2 -2 + -9t + 6t2 = 0 Solving -2 + -9t + 6t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.3333333333 + -1.5t + t2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + -1.5t + 0.3333333333 + t2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + -1.5t + t2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + -1.5t + t2 = 0 + 0.3333333333 -1.5t + t2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 -1.5t + t2 = 0.3333333333 The t term is -1.5t. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5t + 0.5625 + t2 = 0.3333333333 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 0.3333333333 + 0.5625 Combine like terms: 0.3333333333 + 0.5625 = 0.8958333333 0.5625 + -1.5t + t2 = 0.8958333333 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 0.8958333333 Calculate the square root of the right side: 0.946484724 Break this problem into two subproblems by setting (t + -0.75) equal to 0.946484724 and -0.946484724.Subproblem 1
t + -0.75 = 0.946484724 Simplifying t + -0.75 = 0.946484724 Reorder the terms: -0.75 + t = 0.946484724 Solving -0.75 + t = 0.946484724 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = 0.946484724 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 0.946484724 + 0.75 t = 0.946484724 + 0.75 Combine like terms: 0.946484724 + 0.75 = 1.696484724 t = 1.696484724 Simplifying t = 1.696484724Subproblem 2
t + -0.75 = -0.946484724 Simplifying t + -0.75 = -0.946484724 Reorder the terms: -0.75 + t = -0.946484724 Solving -0.75 + t = -0.946484724 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = -0.946484724 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -0.946484724 + 0.75 t = -0.946484724 + 0.75 Combine like terms: -0.946484724 + 0.75 = -0.196484724 t = -0.196484724 Simplifying t = -0.196484724Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.696484724, -0.196484724}
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