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Simplifying (8t + -7)(3t + 5) = 5 Reorder the terms: (-7 + 8t)(3t + 5) = 5 Reorder the terms: (-7 + 8t)(5 + 3t) = 5 Multiply (-7 + 8t) * (5 + 3t) (-7(5 + 3t) + 8t * (5 + 3t)) = 5 ((5 * -7 + 3t * -7) + 8t * (5 + 3t)) = 5 ((-35 + -21t) + 8t * (5 + 3t)) = 5 (-35 + -21t + (5 * 8t + 3t * 8t)) = 5 (-35 + -21t + (40t + 24t2)) = 5 Combine like terms: -21t + 40t = 19t (-35 + 19t + 24t2) = 5 Solving -35 + 19t + 24t2 = 5 Solving for variable 't'. Reorder the terms: -35 + -5 + 19t + 24t2 = 5 + -5 Combine like terms: -35 + -5 = -40 -40 + 19t + 24t2 = 5 + -5 Combine like terms: 5 + -5 = 0 -40 + 19t + 24t2 = 0 Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -1.666666667 + 0.7916666667t + t2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + 0.7916666667t + 1.666666667 + t2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + 0.7916666667t + t2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + 0.7916666667t + t2 = 0 + 1.666666667 0.7916666667t + t2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 0.7916666667t + t2 = 1.666666667 The t term is 0.7916666667t. Take half its coefficient (0.3958333334). Square it (0.1566840278) and add it to both sides. Add '0.1566840278' to each side of the equation. 0.7916666667t + 0.1566840278 + t2 = 1.666666667 + 0.1566840278 Reorder the terms: 0.1566840278 + 0.7916666667t + t2 = 1.666666667 + 0.1566840278 Combine like terms: 1.666666667 + 0.1566840278 = 1.8233506948 0.1566840278 + 0.7916666667t + t2 = 1.8233506948 Factor a perfect square on the left side: (t + 0.3958333334)(t + 0.3958333334) = 1.8233506948 Calculate the square root of the right side: 1.350315035 Break this problem into two subproblems by setting (t + 0.3958333334) equal to 1.350315035 and -1.350315035.Subproblem 1
t + 0.3958333334 = 1.350315035 Simplifying t + 0.3958333334 = 1.350315035 Reorder the terms: 0.3958333334 + t = 1.350315035 Solving 0.3958333334 + t = 1.350315035 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.3958333334' to each side of the equation. 0.3958333334 + -0.3958333334 + t = 1.350315035 + -0.3958333334 Combine like terms: 0.3958333334 + -0.3958333334 = 0.0000000000 0.0000000000 + t = 1.350315035 + -0.3958333334 t = 1.350315035 + -0.3958333334 Combine like terms: 1.350315035 + -0.3958333334 = 0.9544817016 t = 0.9544817016 Simplifying t = 0.9544817016Subproblem 2
t + 0.3958333334 = -1.350315035 Simplifying t + 0.3958333334 = -1.350315035 Reorder the terms: 0.3958333334 + t = -1.350315035 Solving 0.3958333334 + t = -1.350315035 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.3958333334' to each side of the equation. 0.3958333334 + -0.3958333334 + t = -1.350315035 + -0.3958333334 Combine like terms: 0.3958333334 + -0.3958333334 = 0.0000000000 0.0000000000 + t = -1.350315035 + -0.3958333334 t = -1.350315035 + -0.3958333334 Combine like terms: -1.350315035 + -0.3958333334 = -1.7461483684 t = -1.7461483684 Simplifying t = -1.7461483684Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.9544817016, -1.7461483684}
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