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Simplifying (4t + -23)(t + 5) = -110 Reorder the terms: (-23 + 4t)(t + 5) = -110 Reorder the terms: (-23 + 4t)(5 + t) = -110 Multiply (-23 + 4t) * (5 + t) (-23(5 + t) + 4t * (5 + t)) = -110 ((5 * -23 + t * -23) + 4t * (5 + t)) = -110 ((-115 + -23t) + 4t * (5 + t)) = -110 (-115 + -23t + (5 * 4t + t * 4t)) = -110 (-115 + -23t + (20t + 4t2)) = -110 Combine like terms: -23t + 20t = -3t (-115 + -3t + 4t2) = -110 Solving -115 + -3t + 4t2 = -110 Solving for variable 't'. Reorder the terms: -115 + 110 + -3t + 4t2 = -110 + 110 Combine like terms: -115 + 110 = -5 -5 + -3t + 4t2 = -110 + 110 Combine like terms: -110 + 110 = 0 -5 + -3t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.25 + -0.75t + t2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + -0.75t + 1.25 + t2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + -0.75t + t2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + -0.75t + t2 = 0 + 1.25 -0.75t + t2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 -0.75t + t2 = 1.25 The t term is -0.75t. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75t + 0.140625 + t2 = 1.25 + 0.140625 Reorder the terms: 0.140625 + -0.75t + t2 = 1.25 + 0.140625 Combine like terms: 1.25 + 0.140625 = 1.390625 0.140625 + -0.75t + t2 = 1.390625 Factor a perfect square on the left side: (t + -0.375)(t + -0.375) = 1.390625 Calculate the square root of the right side: 1.179247642 Break this problem into two subproblems by setting (t + -0.375) equal to 1.179247642 and -1.179247642.Subproblem 1
t + -0.375 = 1.179247642 Simplifying t + -0.375 = 1.179247642 Reorder the terms: -0.375 + t = 1.179247642 Solving -0.375 + t = 1.179247642 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + t = 1.179247642 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = 1.179247642 + 0.375 t = 1.179247642 + 0.375 Combine like terms: 1.179247642 + 0.375 = 1.554247642 t = 1.554247642 Simplifying t = 1.554247642Subproblem 2
t + -0.375 = -1.179247642 Simplifying t + -0.375 = -1.179247642 Reorder the terms: -0.375 + t = -1.179247642 Solving -0.375 + t = -1.179247642 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + t = -1.179247642 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = -1.179247642 + 0.375 t = -1.179247642 + 0.375 Combine like terms: -1.179247642 + 0.375 = -0.804247642 t = -0.804247642 Simplifying t = -0.804247642Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.554247642, -0.804247642}
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