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Simplifying -16t2 + -6t + 500 = 446 Reorder the terms: 500 + -6t + -16t2 = 446 Solving 500 + -6t + -16t2 = 446 Solving for variable 't'. Reorder the terms: 500 + -446 + -6t + -16t2 = 446 + -446 Combine like terms: 500 + -446 = 54 54 + -6t + -16t2 = 446 + -446 Combine like terms: 446 + -446 = 0 54 + -6t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(27 + -3t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(27 + -3t + -8t2)' equal to zero and attempt to solve: Simplifying 27 + -3t + -8t2 = 0 Solving 27 + -3t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -3.375 + 0.375t + t2 = 0 Move the constant term to the right: Add '3.375' to each side of the equation. -3.375 + 0.375t + 3.375 + t2 = 0 + 3.375 Reorder the terms: -3.375 + 3.375 + 0.375t + t2 = 0 + 3.375 Combine like terms: -3.375 + 3.375 = 0.000 0.000 + 0.375t + t2 = 0 + 3.375 0.375t + t2 = 0 + 3.375 Combine like terms: 0 + 3.375 = 3.375 0.375t + t2 = 3.375 The t term is 0.375t. Take half its coefficient (0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. 0.375t + 0.03515625 + t2 = 3.375 + 0.03515625 Reorder the terms: 0.03515625 + 0.375t + t2 = 3.375 + 0.03515625 Combine like terms: 3.375 + 0.03515625 = 3.41015625 0.03515625 + 0.375t + t2 = 3.41015625 Factor a perfect square on the left side: (t + 0.1875)(t + 0.1875) = 3.41015625 Calculate the square root of the right side: 1.846660838 Break this problem into two subproblems by setting (t + 0.1875) equal to 1.846660838 and -1.846660838.Subproblem 1
t + 0.1875 = 1.846660838 Simplifying t + 0.1875 = 1.846660838 Reorder the terms: 0.1875 + t = 1.846660838 Solving 0.1875 + t = 1.846660838 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + t = 1.846660838 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + t = 1.846660838 + -0.1875 t = 1.846660838 + -0.1875 Combine like terms: 1.846660838 + -0.1875 = 1.659160838 t = 1.659160838 Simplifying t = 1.659160838Subproblem 2
t + 0.1875 = -1.846660838 Simplifying t + 0.1875 = -1.846660838 Reorder the terms: 0.1875 + t = -1.846660838 Solving 0.1875 + t = -1.846660838 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + t = -1.846660838 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + t = -1.846660838 + -0.1875 t = -1.846660838 + -0.1875 Combine like terms: -1.846660838 + -0.1875 = -2.034160838 t = -2.034160838 Simplifying t = -2.034160838Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.659160838, -2.034160838}Solution
t = {1.659160838, -2.034160838}
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