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Simplifying 312 = 180t + -16t2 Solving 312 = 180t + -16t2 Solving for variable 't'. Reorder the terms: 312 + -180t + 16t2 = 180t + -180t + -16t2 + 16t2 Combine like terms: 180t + -180t = 0 312 + -180t + 16t2 = 0 + -16t2 + 16t2 312 + -180t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 312 + -180t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(78 + -45t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(78 + -45t + 4t2)' equal to zero and attempt to solve: Simplifying 78 + -45t + 4t2 = 0 Solving 78 + -45t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 19.5 + -11.25t + t2 = 0 Move the constant term to the right: Add '-19.5' to each side of the equation. 19.5 + -11.25t + -19.5 + t2 = 0 + -19.5 Reorder the terms: 19.5 + -19.5 + -11.25t + t2 = 0 + -19.5 Combine like terms: 19.5 + -19.5 = 0.0 0.0 + -11.25t + t2 = 0 + -19.5 -11.25t + t2 = 0 + -19.5 Combine like terms: 0 + -19.5 = -19.5 -11.25t + t2 = -19.5 The t term is -11.25t. Take half its coefficient (-5.625). Square it (31.640625) and add it to both sides. Add '31.640625' to each side of the equation. -11.25t + 31.640625 + t2 = -19.5 + 31.640625 Reorder the terms: 31.640625 + -11.25t + t2 = -19.5 + 31.640625 Combine like terms: -19.5 + 31.640625 = 12.140625 31.640625 + -11.25t + t2 = 12.140625 Factor a perfect square on the left side: (t + -5.625)(t + -5.625) = 12.140625 Calculate the square root of the right side: 3.484339966 Break this problem into two subproblems by setting (t + -5.625) equal to 3.484339966 and -3.484339966.Subproblem 1
t + -5.625 = 3.484339966 Simplifying t + -5.625 = 3.484339966 Reorder the terms: -5.625 + t = 3.484339966 Solving -5.625 + t = 3.484339966 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5.625' to each side of the equation. -5.625 + 5.625 + t = 3.484339966 + 5.625 Combine like terms: -5.625 + 5.625 = 0.000 0.000 + t = 3.484339966 + 5.625 t = 3.484339966 + 5.625 Combine like terms: 3.484339966 + 5.625 = 9.109339966 t = 9.109339966 Simplifying t = 9.109339966Subproblem 2
t + -5.625 = -3.484339966 Simplifying t + -5.625 = -3.484339966 Reorder the terms: -5.625 + t = -3.484339966 Solving -5.625 + t = -3.484339966 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5.625' to each side of the equation. -5.625 + 5.625 + t = -3.484339966 + 5.625 Combine like terms: -5.625 + 5.625 = 0.000 0.000 + t = -3.484339966 + 5.625 t = -3.484339966 + 5.625 Combine like terms: -3.484339966 + 5.625 = 2.140660034 t = 2.140660034 Simplifying t = 2.140660034Solution
The solution to the problem is based on the solutions from the subproblems. t = {9.109339966, 2.140660034}Solution
t = {9.109339966, 2.140660034}
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