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Simplifying 0 = 16t2 + 112t + -4296 Reorder the terms: 0 = -4296 + 112t + 16t2 Solving 0 = -4296 + 112t + 16t2 Solving for variable 't'. Combine like terms: 0 + 4296 = 4296 4296 + -112t + -16t2 = -4296 + 112t + 16t2 + 4296 + -112t + -16t2 Reorder the terms: 4296 + -112t + -16t2 = -4296 + 4296 + 112t + -112t + 16t2 + -16t2 Combine like terms: -4296 + 4296 = 0 4296 + -112t + -16t2 = 0 + 112t + -112t + 16t2 + -16t2 4296 + -112t + -16t2 = 112t + -112t + 16t2 + -16t2 Combine like terms: 112t + -112t = 0 4296 + -112t + -16t2 = 0 + 16t2 + -16t2 4296 + -112t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 4296 + -112t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(537 + -14t + -2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(537 + -14t + -2t2)' equal to zero and attempt to solve: Simplifying 537 + -14t + -2t2 = 0 Solving 537 + -14t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -268.5 + 7t + t2 = 0 Move the constant term to the right: Add '268.5' to each side of the equation. -268.5 + 7t + 268.5 + t2 = 0 + 268.5 Reorder the terms: -268.5 + 268.5 + 7t + t2 = 0 + 268.5 Combine like terms: -268.5 + 268.5 = 0.0 0.0 + 7t + t2 = 0 + 268.5 7t + t2 = 0 + 268.5 Combine like terms: 0 + 268.5 = 268.5 7t + t2 = 268.5 The t term is 7t. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7t + 12.25 + t2 = 268.5 + 12.25 Reorder the terms: 12.25 + 7t + t2 = 268.5 + 12.25 Combine like terms: 268.5 + 12.25 = 280.75 12.25 + 7t + t2 = 280.75 Factor a perfect square on the left side: (t + 3.5)(t + 3.5) = 280.75 Calculate the square root of the right side: 16.75559608 Break this problem into two subproblems by setting (t + 3.5) equal to 16.75559608 and -16.75559608.Subproblem 1
t + 3.5 = 16.75559608 Simplifying t + 3.5 = 16.75559608 Reorder the terms: 3.5 + t = 16.75559608 Solving 3.5 + t = 16.75559608 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + t = 16.75559608 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + t = 16.75559608 + -3.5 t = 16.75559608 + -3.5 Combine like terms: 16.75559608 + -3.5 = 13.25559608 t = 13.25559608 Simplifying t = 13.25559608Subproblem 2
t + 3.5 = -16.75559608 Simplifying t + 3.5 = -16.75559608 Reorder the terms: 3.5 + t = -16.75559608 Solving 3.5 + t = -16.75559608 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + t = -16.75559608 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + t = -16.75559608 + -3.5 t = -16.75559608 + -3.5 Combine like terms: -16.75559608 + -3.5 = -20.25559608 t = -20.25559608 Simplifying t = -20.25559608Solution
The solution to the problem is based on the solutions from the subproblems. t = {13.25559608, -20.25559608}Solution
t = {13.25559608, -20.25559608}
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