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Simplifying 0 = -16t2 + 68t + 672 Reorder the terms: 0 = 672 + 68t + -16t2 Solving 0 = 672 + 68t + -16t2 Solving for variable 't'. Combine like terms: 0 + -672 = -672 -672 + -68t + 16t2 = 672 + 68t + -16t2 + -672 + -68t + 16t2 Reorder the terms: -672 + -68t + 16t2 = 672 + -672 + 68t + -68t + -16t2 + 16t2 Combine like terms: 672 + -672 = 0 -672 + -68t + 16t2 = 0 + 68t + -68t + -16t2 + 16t2 -672 + -68t + 16t2 = 68t + -68t + -16t2 + 16t2 Combine like terms: 68t + -68t = 0 -672 + -68t + 16t2 = 0 + -16t2 + 16t2 -672 + -68t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -672 + -68t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-168 + -17t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-168 + -17t + 4t2)' equal to zero and attempt to solve: Simplifying -168 + -17t + 4t2 = 0 Solving -168 + -17t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -42 + -4.25t + t2 = 0 Move the constant term to the right: Add '42' to each side of the equation. -42 + -4.25t + 42 + t2 = 0 + 42 Reorder the terms: -42 + 42 + -4.25t + t2 = 0 + 42 Combine like terms: -42 + 42 = 0 0 + -4.25t + t2 = 0 + 42 -4.25t + t2 = 0 + 42 Combine like terms: 0 + 42 = 42 -4.25t + t2 = 42 The t term is -4.25t. Take half its coefficient (-2.125). Square it (4.515625) and add it to both sides. Add '4.515625' to each side of the equation. -4.25t + 4.515625 + t2 = 42 + 4.515625 Reorder the terms: 4.515625 + -4.25t + t2 = 42 + 4.515625 Combine like terms: 42 + 4.515625 = 46.515625 4.515625 + -4.25t + t2 = 46.515625 Factor a perfect square on the left side: (t + -2.125)(t + -2.125) = 46.515625 Calculate the square root of the right side: 6.820236433 Break this problem into two subproblems by setting (t + -2.125) equal to 6.820236433 and -6.820236433.Subproblem 1
t + -2.125 = 6.820236433 Simplifying t + -2.125 = 6.820236433 Reorder the terms: -2.125 + t = 6.820236433 Solving -2.125 + t = 6.820236433 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.125' to each side of the equation. -2.125 + 2.125 + t = 6.820236433 + 2.125 Combine like terms: -2.125 + 2.125 = 0.000 0.000 + t = 6.820236433 + 2.125 t = 6.820236433 + 2.125 Combine like terms: 6.820236433 + 2.125 = 8.945236433 t = 8.945236433 Simplifying t = 8.945236433Subproblem 2
t + -2.125 = -6.820236433 Simplifying t + -2.125 = -6.820236433 Reorder the terms: -2.125 + t = -6.820236433 Solving -2.125 + t = -6.820236433 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.125' to each side of the equation. -2.125 + 2.125 + t = -6.820236433 + 2.125 Combine like terms: -2.125 + 2.125 = 0.000 0.000 + t = -6.820236433 + 2.125 t = -6.820236433 + 2.125 Combine like terms: -6.820236433 + 2.125 = -4.695236433 t = -4.695236433 Simplifying t = -4.695236433Solution
The solution to the problem is based on the solutions from the subproblems. t = {8.945236433, -4.695236433}Solution
t = {8.945236433, -4.695236433}
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