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Simplifying -192 = -6t2 + 64t + 192 Reorder the terms: -192 = 192 + 64t + -6t2 Solving -192 = 192 + 64t + -6t2 Solving for variable 't'. Combine like terms: -192 + -192 = -384 -384 + -64t + 6t2 = 192 + 64t + -6t2 + -192 + -64t + 6t2 Reorder the terms: -384 + -64t + 6t2 = 192 + -192 + 64t + -64t + -6t2 + 6t2 Combine like terms: 192 + -192 = 0 -384 + -64t + 6t2 = 0 + 64t + -64t + -6t2 + 6t2 -384 + -64t + 6t2 = 64t + -64t + -6t2 + 6t2 Combine like terms: 64t + -64t = 0 -384 + -64t + 6t2 = 0 + -6t2 + 6t2 -384 + -64t + 6t2 = -6t2 + 6t2 Combine like terms: -6t2 + 6t2 = 0 -384 + -64t + 6t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-192 + -32t + 3t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-192 + -32t + 3t2)' equal to zero and attempt to solve: Simplifying -192 + -32t + 3t2 = 0 Solving -192 + -32t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -64 + -10.66666667t + t2 = 0 Move the constant term to the right: Add '64' to each side of the equation. -64 + -10.66666667t + 64 + t2 = 0 + 64 Reorder the terms: -64 + 64 + -10.66666667t + t2 = 0 + 64 Combine like terms: -64 + 64 = 0 0 + -10.66666667t + t2 = 0 + 64 -10.66666667t + t2 = 0 + 64 Combine like terms: 0 + 64 = 64 -10.66666667t + t2 = 64 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 = 64 + 28.44444446 Reorder the terms: 28.44444446 + -10.66666667t + t2 = 64 + 28.44444446 Combine like terms: 64 + 28.44444446 = 92.44444446 28.44444446 + -10.66666667t + t2 = 92.44444446 Factor a perfect square on the left side: (t + -5.333333335)(t + -5.333333335) = 92.44444446 Calculate the square root of the right side: 9.614803402 Break this problem into two subproblems by setting (t + -5.333333335) equal to 9.614803402 and -9.614803402.Subproblem 1
t + -5.333333335 = 9.614803402 Simplifying t + -5.333333335 = 9.614803402 Reorder the terms: -5.333333335 + t = 9.614803402 Solving -5.333333335 + t = 9.614803402 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 = 9.614803402 + 5.333333335 Combine like terms: -5.333333335 + 5.333333335 = 0.000000000 0.000000000 + t = 9.614803402 + 5.333333335 t = 9.614803402 + 5.333333335 Combine like terms: 9.614803402 + 5.333333335 = 14.948136737 t = 14.948136737 Simplifying t = 14.948136737Subproblem 2
t + -5.333333335 = -9.614803402 Simplifying t + -5.333333335 = -9.614803402 Reorder the terms: -5.333333335 + t = -9.614803402 Solving -5.333333335 + t = -9.614803402 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 = -9.614803402 + 5.333333335 Combine like terms: -5.333333335 + 5.333333335 = 0.000000000 0.000000000 + t = -9.614803402 + 5.333333335 t = -9.614803402 + 5.333333335 Combine like terms: -9.614803402 + 5.333333335 = -4.281470067 t = -4.281470067 Simplifying t = -4.281470067Solution
The solution to the problem is based on the solutions from the subproblems. t = {14.948136737, -4.281470067}Solution
t = {14.948136737, -4.281470067}
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