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Simplifying 8.5t2 + 5t = 14 Reorder the terms: 5t + 8.5t2 = 14 Solving 5t + 8.5t2 = 14 Solving for variable 't'. Reorder the terms: -14 + 5t + 8.5t2 = 14 + -14 Combine like terms: 14 + -14 = 0 -14 + 5t + 8.5t2 = 0 Begin completing the square. Divide all terms by 8.5 the coefficient of the squared term: Divide each side by '8.5'. -1.647058824 + 0.5882352941t + t2 = 0 Move the constant term to the right: Add '1.647058824' to each side of the equation. -1.647058824 + 0.5882352941t + 1.647058824 + t2 = 0 + 1.647058824 Reorder the terms: -1.647058824 + 1.647058824 + 0.5882352941t + t2 = 0 + 1.647058824 Combine like terms: -1.647058824 + 1.647058824 = 0.000000000 0.000000000 + 0.5882352941t + t2 = 0 + 1.647058824 0.5882352941t + t2 = 0 + 1.647058824 Combine like terms: 0 + 1.647058824 = 1.647058824 0.5882352941t + t2 = 1.647058824 The t term is 0.5882352941t. Take half its coefficient (0.2941176471). Square it (0.08650519034) and add it to both sides. Add '0.08650519034' to each side of the equation. 0.5882352941t + 0.08650519034 + t2 = 1.647058824 + 0.08650519034 Reorder the terms: 0.08650519034 + 0.5882352941t + t2 = 1.647058824 + 0.08650519034 Combine like terms: 1.647058824 + 0.08650519034 = 1.73356401434 0.08650519034 + 0.5882352941t + t2 = 1.73356401434 Factor a perfect square on the left side: (t + 0.2941176471)(t + 0.2941176471) = 1.73356401434 Calculate the square root of the right side: 1.316648782 Break this problem into two subproblems by setting (t + 0.2941176471) equal to 1.316648782 and -1.316648782.Subproblem 1
t + 0.2941176471 = 1.316648782 Simplifying t + 0.2941176471 = 1.316648782 Reorder the terms: 0.2941176471 + t = 1.316648782 Solving 0.2941176471 + t = 1.316648782 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.2941176471' to each side of the equation. 0.2941176471 + -0.2941176471 + t = 1.316648782 + -0.2941176471 Combine like terms: 0.2941176471 + -0.2941176471 = 0.0000000000 0.0000000000 + t = 1.316648782 + -0.2941176471 t = 1.316648782 + -0.2941176471 Combine like terms: 1.316648782 + -0.2941176471 = 1.0225311349 t = 1.0225311349 Simplifying t = 1.0225311349Subproblem 2
t + 0.2941176471 = -1.316648782 Simplifying t + 0.2941176471 = -1.316648782 Reorder the terms: 0.2941176471 + t = -1.316648782 Solving 0.2941176471 + t = -1.316648782 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.2941176471' to each side of the equation. 0.2941176471 + -0.2941176471 + t = -1.316648782 + -0.2941176471 Combine like terms: 0.2941176471 + -0.2941176471 = 0.0000000000 0.0000000000 + t = -1.316648782 + -0.2941176471 t = -1.316648782 + -0.2941176471 Combine like terms: -1.316648782 + -0.2941176471 = -1.6107664291 t = -1.6107664291 Simplifying t = -1.6107664291Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.0225311349, -1.6107664291}
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