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Simplifying 21 = 7 + 38t + -16t2 Solving 21 = 7 + 38t + -16t2 Solving for variable 't'. Combine like terms: 21 + -7 = 14 14 + -38t + 16t2 = 7 + 38t + -16t2 + -7 + -38t + 16t2 Reorder the terms: 14 + -38t + 16t2 = 7 + -7 + 38t + -38t + -16t2 + 16t2 Combine like terms: 7 + -7 = 0 14 + -38t + 16t2 = 0 + 38t + -38t + -16t2 + 16t2 14 + -38t + 16t2 = 38t + -38t + -16t2 + 16t2 Combine like terms: 38t + -38t = 0 14 + -38t + 16t2 = 0 + -16t2 + 16t2 14 + -38t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 14 + -38t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(7 + -19t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(7 + -19t + 8t2)' equal to zero and attempt to solve: Simplifying 7 + -19t + 8t2 = 0 Solving 7 + -19t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.875 + -2.375t + t2 = 0 Move the constant term to the right: Add '-0.875' to each side of the equation. 0.875 + -2.375t + -0.875 + t2 = 0 + -0.875 Reorder the terms: 0.875 + -0.875 + -2.375t + t2 = 0 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + -2.375t + t2 = 0 + -0.875 -2.375t + t2 = 0 + -0.875 Combine like terms: 0 + -0.875 = -0.875 -2.375t + t2 = -0.875 The t term is -2.375t. Take half its coefficient (-1.1875). Square it (1.41015625) and add it to both sides. Add '1.41015625' to each side of the equation. -2.375t + 1.41015625 + t2 = -0.875 + 1.41015625 Reorder the terms: 1.41015625 + -2.375t + t2 = -0.875 + 1.41015625 Combine like terms: -0.875 + 1.41015625 = 0.53515625 1.41015625 + -2.375t + t2 = 0.53515625 Factor a perfect square on the left side: (t + -1.1875)(t + -1.1875) = 0.53515625 Calculate the square root of the right side: 0.731543744 Break this problem into two subproblems by setting (t + -1.1875) equal to 0.731543744 and -0.731543744.Subproblem 1
t + -1.1875 = 0.731543744 Simplifying t + -1.1875 = 0.731543744 Reorder the terms: -1.1875 + t = 0.731543744 Solving -1.1875 + t = 0.731543744 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + t = 0.731543744 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + t = 0.731543744 + 1.1875 t = 0.731543744 + 1.1875 Combine like terms: 0.731543744 + 1.1875 = 1.919043744 t = 1.919043744 Simplifying t = 1.919043744Subproblem 2
t + -1.1875 = -0.731543744 Simplifying t + -1.1875 = -0.731543744 Reorder the terms: -1.1875 + t = -0.731543744 Solving -1.1875 + t = -0.731543744 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + t = -0.731543744 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + t = -0.731543744 + 1.1875 t = -0.731543744 + 1.1875 Combine like terms: -0.731543744 + 1.1875 = 0.455956256 t = 0.455956256 Simplifying t = 0.455956256Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.919043744, 0.455956256}Solution
t = {1.919043744, 0.455956256}
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