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Simplifying -16t2 + 12t + 1000 = 0 Reorder the terms: 1000 + 12t + -16t2 = 0 Solving 1000 + 12t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '4'. 4(250 + 3t + -4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(250 + 3t + -4t2)' equal to zero and attempt to solve: Simplifying 250 + 3t + -4t2 = 0 Solving 250 + 3t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -62.5 + -0.75t + t2 = 0 Move the constant term to the right: Add '62.5' to each side of the equation. -62.5 + -0.75t + 62.5 + t2 = 0 + 62.5 Reorder the terms: -62.5 + 62.5 + -0.75t + t2 = 0 + 62.5 Combine like terms: -62.5 + 62.5 = 0.0 0.0 + -0.75t + t2 = 0 + 62.5 -0.75t + t2 = 0 + 62.5 Combine like terms: 0 + 62.5 = 62.5 -0.75t + t2 = 62.5 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 = 62.5 + 0.140625 Reorder the terms: 0.140625 + -0.75t + t2 = 62.5 + 0.140625 Combine like terms: 62.5 + 0.140625 = 62.640625 0.140625 + -0.75t + t2 = 62.640625 Factor a perfect square on the left side: (t + -0.375)(t + -0.375) = 62.640625 Calculate the square root of the right side: 7.914583059 Break this problem into two subproblems by setting (t + -0.375) equal to 7.914583059 and -7.914583059.Subproblem 1
t + -0.375 = 7.914583059 Simplifying t + -0.375 = 7.914583059 Reorder the terms: -0.375 + t = 7.914583059 Solving -0.375 + t = 7.914583059 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 = 7.914583059 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = 7.914583059 + 0.375 t = 7.914583059 + 0.375 Combine like terms: 7.914583059 + 0.375 = 8.289583059 t = 8.289583059 Simplifying t = 8.289583059Subproblem 2
t + -0.375 = -7.914583059 Simplifying t + -0.375 = -7.914583059 Reorder the terms: -0.375 + t = -7.914583059 Solving -0.375 + t = -7.914583059 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 = -7.914583059 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = -7.914583059 + 0.375 t = -7.914583059 + 0.375 Combine like terms: -7.914583059 + 0.375 = -7.539583059 t = -7.539583059 Simplifying t = -7.539583059Solution
The solution to the problem is based on the solutions from the subproblems. t = {8.289583059, -7.539583059}Solution
t = {8.289583059, -7.539583059}
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