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Simplifying -5(t + -4) * t = 5 + 1 Reorder the terms: -5(-4 + t) * t = 5 + 1 Reorder the terms for easier multiplication: -5t(-4 + t) = 5 + 1 (-4 * -5t + t * -5t) = 5 + 1 (20t + -5t2) = 5 + 1 Combine like terms: 5 + 1 = 6 20t + -5t2 = 6 Solving 20t + -5t2 = 6 Solving for variable 't'. Reorder the terms: -6 + 20t + -5t2 = 6 + -6 Combine like terms: 6 + -6 = 0 -6 + 20t + -5t2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. 1.2 + -4t + t2 = 0 Move the constant term to the right: Add '-1.2' to each side of the equation. 1.2 + -4t + -1.2 + t2 = 0 + -1.2 Reorder the terms: 1.2 + -1.2 + -4t + t2 = 0 + -1.2 Combine like terms: 1.2 + -1.2 = 0.0 0.0 + -4t + t2 = 0 + -1.2 -4t + t2 = 0 + -1.2 Combine like terms: 0 + -1.2 = -1.2 -4t + t2 = -1.2 The t term is -4t. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4t + 4 + t2 = -1.2 + 4 Reorder the terms: 4 + -4t + t2 = -1.2 + 4 Combine like terms: -1.2 + 4 = 2.8 4 + -4t + t2 = 2.8 Factor a perfect square on the left side: (t + -2)(t + -2) = 2.8 Calculate the square root of the right side: 1.673320053 Break this problem into two subproblems by setting (t + -2) equal to 1.673320053 and -1.673320053.Subproblem 1
t + -2 = 1.673320053 Simplifying t + -2 = 1.673320053 Reorder the terms: -2 + t = 1.673320053 Solving -2 + t = 1.673320053 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = 1.673320053 + 2 Combine like terms: -2 + 2 = 0 0 + t = 1.673320053 + 2 t = 1.673320053 + 2 Combine like terms: 1.673320053 + 2 = 3.673320053 t = 3.673320053 Simplifying t = 3.673320053Subproblem 2
t + -2 = -1.673320053 Simplifying t + -2 = -1.673320053 Reorder the terms: -2 + t = -1.673320053 Solving -2 + t = -1.673320053 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = -1.673320053 + 2 Combine like terms: -2 + 2 = 0 0 + t = -1.673320053 + 2 t = -1.673320053 + 2 Combine like terms: -1.673320053 + 2 = 0.326679947 t = 0.326679947 Simplifying t = 0.326679947Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.673320053, 0.326679947}
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