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Simplifying 12 + -2t(t + -3) = -8t Reorder the terms: 12 + -2t(-3 + t) = -8t 12 + (-3 * -2t + t * -2t) = -8t 12 + (6t + -2t2) = -8t Solving 12 + 6t + -2t2 = -8t Solving for variable 't'. Reorder the terms: 12 + 6t + 8t + -2t2 = -8t + 8t Combine like terms: 6t + 8t = 14t 12 + 14t + -2t2 = -8t + 8t Combine like terms: -8t + 8t = 0 12 + 14t + -2t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(6 + 7t + -1t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(6 + 7t + -1t2)' equal to zero and attempt to solve: Simplifying 6 + 7t + -1t2 = 0 Solving 6 + 7t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -6 + -7t + t2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + -7t + 6 + t2 = 0 + 6 Reorder the terms: -6 + 6 + -7t + t2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -7t + t2 = 0 + 6 -7t + t2 = 0 + 6 Combine like terms: 0 + 6 = 6 -7t + t2 = 6 The t term is -7t. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7t + 12.25 + t2 = 6 + 12.25 Reorder the terms: 12.25 + -7t + t2 = 6 + 12.25 Combine like terms: 6 + 12.25 = 18.25 12.25 + -7t + t2 = 18.25 Factor a perfect square on the left side: (t + -3.5)(t + -3.5) = 18.25 Calculate the square root of the right side: 4.272001873 Break this problem into two subproblems by setting (t + -3.5) equal to 4.272001873 and -4.272001873.Subproblem 1
t + -3.5 = 4.272001873 Simplifying t + -3.5 = 4.272001873 Reorder the terms: -3.5 + t = 4.272001873 Solving -3.5 + t = 4.272001873 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + t = 4.272001873 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + t = 4.272001873 + 3.5 t = 4.272001873 + 3.5 Combine like terms: 4.272001873 + 3.5 = 7.772001873 t = 7.772001873 Simplifying t = 7.772001873Subproblem 2
t + -3.5 = -4.272001873 Simplifying t + -3.5 = -4.272001873 Reorder the terms: -3.5 + t = -4.272001873 Solving -3.5 + t = -4.272001873 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + t = -4.272001873 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + t = -4.272001873 + 3.5 t = -4.272001873 + 3.5 Combine like terms: -4.272001873 + 3.5 = -0.772001873 t = -0.772001873 Simplifying t = -0.772001873Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.772001873, -0.772001873}Solution
t = {7.772001873, -0.772001873}
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