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Simplifying 4t2 + 23t + 6 = 0 Reorder the terms: 6 + 23t + 4t2 = 0 Solving 6 + 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'. 1.5 + 5.75t + t2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + 5.75t + -1.5 + t2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + 5.75t + t2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + 5.75t + t2 = 0 + -1.5 5.75t + t2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 5.75t + t2 = -1.5 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 = -1.5 + 8.265625 Reorder the terms: 8.265625 + 5.75t + t2 = -1.5 + 8.265625 Combine like terms: -1.5 + 8.265625 = 6.765625 8.265625 + 5.75t + t2 = 6.765625 Factor a perfect square on the left side: (t + 2.875)(t + 2.875) = 6.765625 Calculate the square root of the right side: 2.601081506 Break this problem into two subproblems by setting (t + 2.875) equal to 2.601081506 and -2.601081506.Subproblem 1
t + 2.875 = 2.601081506 Simplifying t + 2.875 = 2.601081506 Reorder the terms: 2.875 + t = 2.601081506 Solving 2.875 + t = 2.601081506 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 = 2.601081506 + -2.875 Combine like terms: 2.875 + -2.875 = 0.000 0.000 + t = 2.601081506 + -2.875 t = 2.601081506 + -2.875 Combine like terms: 2.601081506 + -2.875 = -0.273918494 t = -0.273918494 Simplifying t = -0.273918494Subproblem 2
t + 2.875 = -2.601081506 Simplifying t + 2.875 = -2.601081506 Reorder the terms: 2.875 + t = -2.601081506 Solving 2.875 + t = -2.601081506 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 = -2.601081506 + -2.875 Combine like terms: 2.875 + -2.875 = 0.000 0.000 + t = -2.601081506 + -2.875 t = -2.601081506 + -2.875 Combine like terms: -2.601081506 + -2.875 = -5.476081506 t = -5.476081506 Simplifying t = -5.476081506Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.273918494, -5.476081506}
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