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Simplifying 6t2 + -5.4 + -3.5 + 2.5t2 + 3t = 0 Reorder the terms: -5.4 + -3.5 + 3t + 6t2 + 2.5t2 = 0 Combine like terms: -5.4 + -3.5 = -8.9 -8.9 + 3t + 6t2 + 2.5t2 = 0 Combine like terms: 6t2 + 2.5t2 = 8.5t2 -8.9 + 3t + 8.5t2 = 0 Solving -8.9 + 3t + 8.5t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 8.5 the coefficient of the squared term: Divide each side by '8.5'. -1.047058824 + 0.3529411765t + t2 = 0 Move the constant term to the right: Add '1.047058824' to each side of the equation. -1.047058824 + 0.3529411765t + 1.047058824 + t2 = 0 + 1.047058824 Reorder the terms: -1.047058824 + 1.047058824 + 0.3529411765t + t2 = 0 + 1.047058824 Combine like terms: -1.047058824 + 1.047058824 = 0.000000000 0.000000000 + 0.3529411765t + t2 = 0 + 1.047058824 0.3529411765t + t2 = 0 + 1.047058824 Combine like terms: 0 + 1.047058824 = 1.047058824 0.3529411765t + t2 = 1.047058824 The t term is 0.3529411765t. Take half its coefficient (0.1764705883). Square it (0.03114186853) and add it to both sides. Add '0.03114186853' to each side of the equation. 0.3529411765t + 0.03114186853 + t2 = 1.047058824 + 0.03114186853 Reorder the terms: 0.03114186853 + 0.3529411765t + t2 = 1.047058824 + 0.03114186853 Combine like terms: 1.047058824 + 0.03114186853 = 1.07820069253 0.03114186853 + 0.3529411765t + t2 = 1.07820069253 Factor a perfect square on the left side: (t + 0.1764705883)(t + 0.1764705883) = 1.07820069253 Calculate the square root of the right side: 1.038364431 Break this problem into two subproblems by setting (t + 0.1764705883) equal to 1.038364431 and -1.038364431.Subproblem 1
t + 0.1764705883 = 1.038364431 Simplifying t + 0.1764705883 = 1.038364431 Reorder the terms: 0.1764705883 + t = 1.038364431 Solving 0.1764705883 + t = 1.038364431 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.1764705883' to each side of the equation. 0.1764705883 + -0.1764705883 + t = 1.038364431 + -0.1764705883 Combine like terms: 0.1764705883 + -0.1764705883 = 0.0000000000 0.0000000000 + t = 1.038364431 + -0.1764705883 t = 1.038364431 + -0.1764705883 Combine like terms: 1.038364431 + -0.1764705883 = 0.8618938427 t = 0.8618938427 Simplifying t = 0.8618938427Subproblem 2
t + 0.1764705883 = -1.038364431 Simplifying t + 0.1764705883 = -1.038364431 Reorder the terms: 0.1764705883 + t = -1.038364431 Solving 0.1764705883 + t = -1.038364431 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.1764705883' to each side of the equation. 0.1764705883 + -0.1764705883 + t = -1.038364431 + -0.1764705883 Combine like terms: 0.1764705883 + -0.1764705883 = 0.0000000000 0.0000000000 + t = -1.038364431 + -0.1764705883 t = -1.038364431 + -0.1764705883 Combine like terms: -1.038364431 + -0.1764705883 = -1.2148350193 t = -1.2148350193 Simplifying t = -1.2148350193Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.8618938427, -1.2148350193}
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