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Simplifying 23t2 + -54t + -84 = 0 Reorder the terms: -84 + -54t + 23t2 = 0 Solving -84 + -54t + 23t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 23 the coefficient of the squared term: Divide each side by '23'. -3.652173913 + -2.347826087t + t2 = 0 Move the constant term to the right: Add '3.652173913' to each side of the equation. -3.652173913 + -2.347826087t + 3.652173913 + t2 = 0 + 3.652173913 Reorder the terms: -3.652173913 + 3.652173913 + -2.347826087t + t2 = 0 + 3.652173913 Combine like terms: -3.652173913 + 3.652173913 = 0.000000000 0.000000000 + -2.347826087t + t2 = 0 + 3.652173913 -2.347826087t + t2 = 0 + 3.652173913 Combine like terms: 0 + 3.652173913 = 3.652173913 -2.347826087t + t2 = 3.652173913 The t term is -2.347826087t. Take half its coefficient (-1.173913044). Square it (1.378071835) and add it to both sides. Add '1.378071835' to each side of the equation. -2.347826087t + 1.378071835 + t2 = 3.652173913 + 1.378071835 Reorder the terms: 1.378071835 + -2.347826087t + t2 = 3.652173913 + 1.378071835 Combine like terms: 3.652173913 + 1.378071835 = 5.030245748 1.378071835 + -2.347826087t + t2 = 5.030245748 Factor a perfect square on the left side: (t + -1.173913044)(t + -1.173913044) = 5.030245748 Calculate the square root of the right side: 2.242820935 Break this problem into two subproblems by setting (t + -1.173913044) equal to 2.242820935 and -2.242820935.Subproblem 1
t + -1.173913044 = 2.242820935 Simplifying t + -1.173913044 = 2.242820935 Reorder the terms: -1.173913044 + t = 2.242820935 Solving -1.173913044 + t = 2.242820935 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.173913044' to each side of the equation. -1.173913044 + 1.173913044 + t = 2.242820935 + 1.173913044 Combine like terms: -1.173913044 + 1.173913044 = 0.000000000 0.000000000 + t = 2.242820935 + 1.173913044 t = 2.242820935 + 1.173913044 Combine like terms: 2.242820935 + 1.173913044 = 3.416733979 t = 3.416733979 Simplifying t = 3.416733979Subproblem 2
t + -1.173913044 = -2.242820935 Simplifying t + -1.173913044 = -2.242820935 Reorder the terms: -1.173913044 + t = -2.242820935 Solving -1.173913044 + t = -2.242820935 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.173913044' to each side of the equation. -1.173913044 + 1.173913044 + t = -2.242820935 + 1.173913044 Combine like terms: -1.173913044 + 1.173913044 = 0.000000000 0.000000000 + t = -2.242820935 + 1.173913044 t = -2.242820935 + 1.173913044 Combine like terms: -2.242820935 + 1.173913044 = -1.068907891 t = -1.068907891 Simplifying t = -1.068907891Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.416733979, -1.068907891}
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