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Simplifying 6 = 4t2 + 3t + 2 Reorder the terms: 6 = 2 + 3t + 4t2 Solving 6 = 2 + 3t + 4t2 Solving for variable 't'. Combine like terms: 6 + -2 = 4 4 + -3t + -4t2 = 2 + 3t + 4t2 + -2 + -3t + -4t2 Reorder the terms: 4 + -3t + -4t2 = 2 + -2 + 3t + -3t + 4t2 + -4t2 Combine like terms: 2 + -2 = 0 4 + -3t + -4t2 = 0 + 3t + -3t + 4t2 + -4t2 4 + -3t + -4t2 = 3t + -3t + 4t2 + -4t2 Combine like terms: 3t + -3t = 0 4 + -3t + -4t2 = 0 + 4t2 + -4t2 4 + -3t + -4t2 = 4t2 + -4t2 Combine like terms: 4t2 + -4t2 = 0 4 + -3t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -1 + 0.75t + t2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 0.75t + 1 + t2 = 0 + 1 Reorder the terms: -1 + 1 + 0.75t + t2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 0.75t + t2 = 0 + 1 0.75t + t2 = 0 + 1 Combine like terms: 0 + 1 = 1 0.75t + t2 = 1 The t term is 0.75t. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75t + 0.140625 + t2 = 1 + 0.140625 Reorder the terms: 0.140625 + 0.75t + t2 = 1 + 0.140625 Combine like terms: 1 + 0.140625 = 1.140625 0.140625 + 0.75t + t2 = 1.140625 Factor a perfect square on the left side: (t + 0.375)(t + 0.375) = 1.140625 Calculate the square root of the right side: 1.068000468 Break this problem into two subproblems by setting (t + 0.375) equal to 1.068000468 and -1.068000468.Subproblem 1
t + 0.375 = 1.068000468 Simplifying t + 0.375 = 1.068000468 Reorder the terms: 0.375 + t = 1.068000468 Solving 0.375 + t = 1.068000468 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + t = 1.068000468 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + t = 1.068000468 + -0.375 t = 1.068000468 + -0.375 Combine like terms: 1.068000468 + -0.375 = 0.693000468 t = 0.693000468 Simplifying t = 0.693000468Subproblem 2
t + 0.375 = -1.068000468 Simplifying t + 0.375 = -1.068000468 Reorder the terms: 0.375 + t = -1.068000468 Solving 0.375 + t = -1.068000468 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + t = -1.068000468 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + t = -1.068000468 + -0.375 t = -1.068000468 + -0.375 Combine like terms: -1.068000468 + -0.375 = -1.443000468 t = -1.443000468 Simplifying t = -1.443000468Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.693000468, -1.443000468}
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