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Simplifying 214 = 16t2 + 14t + 0 Reorder the terms: 214 = 0 + 14t + 16t2 Remove the zero: 214 = 14t + 16t2 Solving 214 = 14t + 16t2 Solving for variable 't'. Reorder the terms: 214 + -14t + -16t2 = 14t + -14t + 16t2 + -16t2 Combine like terms: 14t + -14t = 0 214 + -14t + -16t2 = 0 + 16t2 + -16t2 214 + -14t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 214 + -14t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(107 + -7t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(107 + -7t + -8t2)' equal to zero and attempt to solve: Simplifying 107 + -7t + -8t2 = 0 Solving 107 + -7t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -13.375 + 0.875t + t2 = 0 Move the constant term to the right: Add '13.375' to each side of the equation. -13.375 + 0.875t + 13.375 + t2 = 0 + 13.375 Reorder the terms: -13.375 + 13.375 + 0.875t + t2 = 0 + 13.375 Combine like terms: -13.375 + 13.375 = 0.000 0.000 + 0.875t + t2 = 0 + 13.375 0.875t + t2 = 0 + 13.375 Combine like terms: 0 + 13.375 = 13.375 0.875t + t2 = 13.375 The t term is 0.875t. Take half its coefficient (0.4375). Square it (0.19140625) and add it to both sides. Add '0.19140625' to each side of the equation. 0.875t + 0.19140625 + t2 = 13.375 + 0.19140625 Reorder the terms: 0.19140625 + 0.875t + t2 = 13.375 + 0.19140625 Combine like terms: 13.375 + 0.19140625 = 13.56640625 0.19140625 + 0.875t + t2 = 13.56640625 Factor a perfect square on the left side: (t + 0.4375)(t + 0.4375) = 13.56640625 Calculate the square root of the right side: 3.683260275 Break this problem into two subproblems by setting (t + 0.4375) equal to 3.683260275 and -3.683260275.Subproblem 1
t + 0.4375 = 3.683260275 Simplifying t + 0.4375 = 3.683260275 Reorder the terms: 0.4375 + t = 3.683260275 Solving 0.4375 + t = 3.683260275 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.4375' to each side of the equation. 0.4375 + -0.4375 + t = 3.683260275 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + t = 3.683260275 + -0.4375 t = 3.683260275 + -0.4375 Combine like terms: 3.683260275 + -0.4375 = 3.245760275 t = 3.245760275 Simplifying t = 3.245760275Subproblem 2
t + 0.4375 = -3.683260275 Simplifying t + 0.4375 = -3.683260275 Reorder the terms: 0.4375 + t = -3.683260275 Solving 0.4375 + t = -3.683260275 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.4375' to each side of the equation. 0.4375 + -0.4375 + t = -3.683260275 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + t = -3.683260275 + -0.4375 t = -3.683260275 + -0.4375 Combine like terms: -3.683260275 + -0.4375 = -4.120760275 t = -4.120760275 Simplifying t = -4.120760275Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.245760275, -4.120760275}Solution
t = {3.245760275, -4.120760275}
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