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Simplifying -30t = -16t2 + 44t + 6 Reorder the terms: -30t = 6 + 44t + -16t2 Solving -30t = 6 + 44t + -16t2 Solving for variable 't'. Reorder the terms: -6 + -30t + -44t + 16t2 = 6 + 44t + -16t2 + -6 + -44t + 16t2 Combine like terms: -30t + -44t = -74t -6 + -74t + 16t2 = 6 + 44t + -16t2 + -6 + -44t + 16t2 Reorder the terms: -6 + -74t + 16t2 = 6 + -6 + 44t + -44t + -16t2 + 16t2 Combine like terms: 6 + -6 = 0 -6 + -74t + 16t2 = 0 + 44t + -44t + -16t2 + 16t2 -6 + -74t + 16t2 = 44t + -44t + -16t2 + 16t2 Combine like terms: 44t + -44t = 0 -6 + -74t + 16t2 = 0 + -16t2 + 16t2 -6 + -74t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -6 + -74t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + -37t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + -37t + 8t2)' equal to zero and attempt to solve: Simplifying -3 + -37t + 8t2 = 0 Solving -3 + -37t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.375 + -4.625t + t2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + -4.625t + 0.375 + t2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + -4.625t + t2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + -4.625t + t2 = 0 + 0.375 -4.625t + t2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 -4.625t + t2 = 0.375 The t term is -4.625t. Take half its coefficient (-2.3125). Square it (5.34765625) and add it to both sides. Add '5.34765625' to each side of the equation. -4.625t + 5.34765625 + t2 = 0.375 + 5.34765625 Reorder the terms: 5.34765625 + -4.625t + t2 = 0.375 + 5.34765625 Combine like terms: 0.375 + 5.34765625 = 5.72265625 5.34765625 + -4.625t + t2 = 5.72265625 Factor a perfect square on the left side: (t + -2.3125)(t + -2.3125) = 5.72265625 Calculate the square root of the right side: 2.392207401 Break this problem into two subproblems by setting (t + -2.3125) equal to 2.392207401 and -2.392207401.Subproblem 1
t + -2.3125 = 2.392207401 Simplifying t + -2.3125 = 2.392207401 Reorder the terms: -2.3125 + t = 2.392207401 Solving -2.3125 + t = 2.392207401 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.3125' to each side of the equation. -2.3125 + 2.3125 + t = 2.392207401 + 2.3125 Combine like terms: -2.3125 + 2.3125 = 0.0000 0.0000 + t = 2.392207401 + 2.3125 t = 2.392207401 + 2.3125 Combine like terms: 2.392207401 + 2.3125 = 4.704707401 t = 4.704707401 Simplifying t = 4.704707401Subproblem 2
t + -2.3125 = -2.392207401 Simplifying t + -2.3125 = -2.392207401 Reorder the terms: -2.3125 + t = -2.392207401 Solving -2.3125 + t = -2.392207401 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.3125' to each side of the equation. -2.3125 + 2.3125 + t = -2.392207401 + 2.3125 Combine like terms: -2.3125 + 2.3125 = 0.0000 0.0000 + t = -2.392207401 + 2.3125 t = -2.392207401 + 2.3125 Combine like terms: -2.392207401 + 2.3125 = -0.079707401 t = -0.079707401 Simplifying t = -0.079707401Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.704707401, -0.079707401}Solution
t = {4.704707401, -0.079707401}
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