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Simplifying 2(3t + -8) * 4t = 109 Reorder the terms: 2(-8 + 3t) * 4t = 109 Reorder the terms for easier multiplication: 2 * 4t(-8 + 3t) = 109 Multiply 2 * 4 8t(-8 + 3t) = 109 (-8 * 8t + 3t * 8t) = 109 (-64t + 24t2) = 109 Solving -64t + 24t2 = 109 Solving for variable 't'. Reorder the terms: -109 + -64t + 24t2 = 109 + -109 Combine like terms: 109 + -109 = 0 -109 + -64t + 24t2 = 0 Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -4.541666667 + -2.666666667t + t2 = 0 Move the constant term to the right: Add '4.541666667' to each side of the equation. -4.541666667 + -2.666666667t + 4.541666667 + t2 = 0 + 4.541666667 Reorder the terms: -4.541666667 + 4.541666667 + -2.666666667t + t2 = 0 + 4.541666667 Combine like terms: -4.541666667 + 4.541666667 = 0.000000000 0.000000000 + -2.666666667t + t2 = 0 + 4.541666667 -2.666666667t + t2 = 0 + 4.541666667 Combine like terms: 0 + 4.541666667 = 4.541666667 -2.666666667t + t2 = 4.541666667 The t term is -2.666666667t. Take half its coefficient (-1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. -2.666666667t + 1.777777780 + t2 = 4.541666667 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667t + t2 = 4.541666667 + 1.777777780 Combine like terms: 4.541666667 + 1.777777780 = 6.319444447 1.777777780 + -2.666666667t + t2 = 6.319444447 Factor a perfect square on the left side: (t + -1.333333334)(t + -1.333333334) = 6.319444447 Calculate the square root of the right side: 2.513850522 Break this problem into two subproblems by setting (t + -1.333333334) equal to 2.513850522 and -2.513850522.Subproblem 1
t + -1.333333334 = 2.513850522 Simplifying t + -1.333333334 = 2.513850522 Reorder the terms: -1.333333334 + t = 2.513850522 Solving -1.333333334 + t = 2.513850522 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + t = 2.513850522 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + t = 2.513850522 + 1.333333334 t = 2.513850522 + 1.333333334 Combine like terms: 2.513850522 + 1.333333334 = 3.847183856 t = 3.847183856 Simplifying t = 3.847183856Subproblem 2
t + -1.333333334 = -2.513850522 Simplifying t + -1.333333334 = -2.513850522 Reorder the terms: -1.333333334 + t = -2.513850522 Solving -1.333333334 + t = -2.513850522 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + t = -2.513850522 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + t = -2.513850522 + 1.333333334 t = -2.513850522 + 1.333333334 Combine like terms: -2.513850522 + 1.333333334 = -1.180517188 t = -1.180517188 Simplifying t = -1.180517188Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.847183856, -1.180517188}
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