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Simplifying 50 = -16t2 + -14t + 180 Reorder the terms: 50 = 180 + -14t + -16t2 Solving 50 = 180 + -14t + -16t2 Solving for variable 't'. Combine like terms: 50 + -180 = -130 -130 + 14t + 16t2 = 180 + -14t + -16t2 + -180 + 14t + 16t2 Reorder the terms: -130 + 14t + 16t2 = 180 + -180 + -14t + 14t + -16t2 + 16t2 Combine like terms: 180 + -180 = 0 -130 + 14t + 16t2 = 0 + -14t + 14t + -16t2 + 16t2 -130 + 14t + 16t2 = -14t + 14t + -16t2 + 16t2 Combine like terms: -14t + 14t = 0 -130 + 14t + 16t2 = 0 + -16t2 + 16t2 -130 + 14t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -130 + 14t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-65 + 7t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-65 + 7t + 8t2)' equal to zero and attempt to solve: Simplifying -65 + 7t + 8t2 = 0 Solving -65 + 7t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -8.125 + 0.875t + t2 = 0 Move the constant term to the right: Add '8.125' to each side of the equation. -8.125 + 0.875t + 8.125 + t2 = 0 + 8.125 Reorder the terms: -8.125 + 8.125 + 0.875t + t2 = 0 + 8.125 Combine like terms: -8.125 + 8.125 = 0.000 0.000 + 0.875t + t2 = 0 + 8.125 0.875t + t2 = 0 + 8.125 Combine like terms: 0 + 8.125 = 8.125 0.875t + t2 = 8.125 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 = 8.125 + 0.19140625 Reorder the terms: 0.19140625 + 0.875t + t2 = 8.125 + 0.19140625 Combine like terms: 8.125 + 0.19140625 = 8.31640625 0.19140625 + 0.875t + t2 = 8.31640625 Factor a perfect square on the left side: (t + 0.4375)(t + 0.4375) = 8.31640625 Calculate the square root of the right side: 2.883817999 Break this problem into two subproblems by setting (t + 0.4375) equal to 2.883817999 and -2.883817999.Subproblem 1
t + 0.4375 = 2.883817999 Simplifying t + 0.4375 = 2.883817999 Reorder the terms: 0.4375 + t = 2.883817999 Solving 0.4375 + t = 2.883817999 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 = 2.883817999 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + t = 2.883817999 + -0.4375 t = 2.883817999 + -0.4375 Combine like terms: 2.883817999 + -0.4375 = 2.446317999 t = 2.446317999 Simplifying t = 2.446317999Subproblem 2
t + 0.4375 = -2.883817999 Simplifying t + 0.4375 = -2.883817999 Reorder the terms: 0.4375 + t = -2.883817999 Solving 0.4375 + t = -2.883817999 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 = -2.883817999 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + t = -2.883817999 + -0.4375 t = -2.883817999 + -0.4375 Combine like terms: -2.883817999 + -0.4375 = -3.321317999 t = -3.321317999 Simplifying t = -3.321317999Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.446317999, -3.321317999}Solution
t = {2.446317999, -3.321317999}
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