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Simplifying -6t2 + 64t + 192 = 0 Reorder the terms: 192 + 64t + -6t2 = 0 Solving 192 + 64t + -6t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(96 + 32t + -3t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(96 + 32t + -3t2)' equal to zero and attempt to solve: Simplifying 96 + 32t + -3t2 = 0 Solving 96 + 32t + -3t2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -32 + -10.66666667t + t2 = 0 Move the constant term to the right: Add '32' to each side of the equation. -32 + -10.66666667t + 32 + t2 = 0 + 32 Reorder the terms: -32 + 32 + -10.66666667t + t2 = 0 + 32 Combine like terms: -32 + 32 = 0 0 + -10.66666667t + t2 = 0 + 32 -10.66666667t + t2 = 0 + 32 Combine like terms: 0 + 32 = 32 -10.66666667t + t2 = 32 The t term is -10.66666667t. Take half its coefficient (-5.333333335). Square it (28.44444446) and add it to both sides. Add '28.44444446' to each side of the equation. -10.66666667t + 28.44444446 + t2 = 32 + 28.44444446 Reorder the terms: 28.44444446 + -10.66666667t + t2 = 32 + 28.44444446 Combine like terms: 32 + 28.44444446 = 60.44444446 28.44444446 + -10.66666667t + t2 = 60.44444446 Factor a perfect square on the left side: (t + -5.333333335)(t + -5.333333335) = 60.44444446 Calculate the square root of the right side: 7.774602527 Break this problem into two subproblems by setting (t + -5.333333335) equal to 7.774602527 and -7.774602527.Subproblem 1
t + -5.333333335 = 7.774602527 Simplifying t + -5.333333335 = 7.774602527 Reorder the terms: -5.333333335 + t = 7.774602527 Solving -5.333333335 + t = 7.774602527 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5.333333335' to each side of the equation. -5.333333335 + 5.333333335 + t = 7.774602527 + 5.333333335 Combine like terms: -5.333333335 + 5.333333335 = 0.000000000 0.000000000 + t = 7.774602527 + 5.333333335 t = 7.774602527 + 5.333333335 Combine like terms: 7.774602527 + 5.333333335 = 13.107935862 t = 13.107935862 Simplifying t = 13.107935862Subproblem 2
t + -5.333333335 = -7.774602527 Simplifying t + -5.333333335 = -7.774602527 Reorder the terms: -5.333333335 + t = -7.774602527 Solving -5.333333335 + t = -7.774602527 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5.333333335' to each side of the equation. -5.333333335 + 5.333333335 + t = -7.774602527 + 5.333333335 Combine like terms: -5.333333335 + 5.333333335 = 0.000000000 0.000000000 + t = -7.774602527 + 5.333333335 t = -7.774602527 + 5.333333335 Combine like terms: -7.774602527 + 5.333333335 = -2.441269192 t = -2.441269192 Simplifying t = -2.441269192Solution
The solution to the problem is based on the solutions from the subproblems. t = {13.107935862, -2.441269192}Solution
t = {13.107935862, -2.441269192}
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