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Simplifying 4t2 + 23t + 8 = 0 Reorder the terms: 8 + 23t + 4t2 = 0 Solving 8 + 23t + 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'. 2 + 5.75t + t2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + 5.75t + -2 + t2 = 0 + -2 Reorder the terms: 2 + -2 + 5.75t + t2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + 5.75t + t2 = 0 + -2 5.75t + t2 = 0 + -2 Combine like terms: 0 + -2 = -2 5.75t + t2 = -2 The t term is 5.75t. Take half its coefficient (2.875). Square it (8.265625) and add it to both sides. Add '8.265625' to each side of the equation. 5.75t + 8.265625 + t2 = -2 + 8.265625 Reorder the terms: 8.265625 + 5.75t + t2 = -2 + 8.265625 Combine like terms: -2 + 8.265625 = 6.265625 8.265625 + 5.75t + t2 = 6.265625 Factor a perfect square on the left side: (t + 2.875)(t + 2.875) = 6.265625 Calculate the square root of the right side: 2.503123049 Break this problem into two subproblems by setting (t + 2.875) equal to 2.503123049 and -2.503123049.Subproblem 1
t + 2.875 = 2.503123049 Simplifying t + 2.875 = 2.503123049 Reorder the terms: 2.875 + t = 2.503123049 Solving 2.875 + t = 2.503123049 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.875' to each side of the equation. 2.875 + -2.875 + t = 2.503123049 + -2.875 Combine like terms: 2.875 + -2.875 = 0.000 0.000 + t = 2.503123049 + -2.875 t = 2.503123049 + -2.875 Combine like terms: 2.503123049 + -2.875 = -0.371876951 t = -0.371876951 Simplifying t = -0.371876951Subproblem 2
t + 2.875 = -2.503123049 Simplifying t + 2.875 = -2.503123049 Reorder the terms: 2.875 + t = -2.503123049 Solving 2.875 + t = -2.503123049 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.875' to each side of the equation. 2.875 + -2.875 + t = -2.503123049 + -2.875 Combine like terms: 2.875 + -2.875 = 0.000 0.000 + t = -2.503123049 + -2.875 t = -2.503123049 + -2.875 Combine like terms: -2.503123049 + -2.875 = -5.378123049 t = -5.378123049 Simplifying t = -5.378123049Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.371876951, -5.378123049}
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