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Simplifying 2(3t + -8) * 4t = 10 Reorder the terms: 2(-8 + 3t) * 4t = 10 Reorder the terms for easier multiplication: 2 * 4t(-8 + 3t) = 10 Multiply 2 * 4 8t(-8 + 3t) = 10 (-8 * 8t + 3t * 8t) = 10 (-64t + 24t2) = 10 Solving -64t + 24t2 = 10 Solving for variable 't'. Reorder the terms: -10 + -64t + 24t2 = 10 + -10 Combine like terms: 10 + -10 = 0 -10 + -64t + 24t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-5 + -32t + 12t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-5 + -32t + 12t2)' equal to zero and attempt to solve: Simplifying -5 + -32t + 12t2 = 0 Solving -5 + -32t + 12t2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.4166666667 + -2.666666667t + t2 = 0 Move the constant term to the right: Add '0.4166666667' to each side of the equation. -0.4166666667 + -2.666666667t + 0.4166666667 + t2 = 0 + 0.4166666667 Reorder the terms: -0.4166666667 + 0.4166666667 + -2.666666667t + t2 = 0 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + -2.666666667t + t2 = 0 + 0.4166666667 -2.666666667t + t2 = 0 + 0.4166666667 Combine like terms: 0 + 0.4166666667 = 0.4166666667 -2.666666667t + t2 = 0.4166666667 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 = 0.4166666667 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667t + t2 = 0.4166666667 + 1.777777780 Combine like terms: 0.4166666667 + 1.777777780 = 2.1944444467 1.777777780 + -2.666666667t + t2 = 2.1944444467 Factor a perfect square on the left side: (t + -1.333333334)(t + -1.333333334) = 2.1944444467 Calculate the square root of the right side: 1.481365737 Break this problem into two subproblems by setting (t + -1.333333334) equal to 1.481365737 and -1.481365737.Subproblem 1
t + -1.333333334 = 1.481365737 Simplifying t + -1.333333334 = 1.481365737 Reorder the terms: -1.333333334 + t = 1.481365737 Solving -1.333333334 + t = 1.481365737 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 = 1.481365737 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + t = 1.481365737 + 1.333333334 t = 1.481365737 + 1.333333334 Combine like terms: 1.481365737 + 1.333333334 = 2.814699071 t = 2.814699071 Simplifying t = 2.814699071Subproblem 2
t + -1.333333334 = -1.481365737 Simplifying t + -1.333333334 = -1.481365737 Reorder the terms: -1.333333334 + t = -1.481365737 Solving -1.333333334 + t = -1.481365737 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 = -1.481365737 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + t = -1.481365737 + 1.333333334 t = -1.481365737 + 1.333333334 Combine like terms: -1.481365737 + 1.333333334 = -0.148032403 t = -0.148032403 Simplifying t = -0.148032403Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.814699071, -0.148032403}Solution
t = {2.814699071, -0.148032403}
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