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Simplifying -16t2 + 46t + 576 = 0 Reorder the terms: 576 + 46t + -16t2 = 0 Solving 576 + 46t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(288 + 23t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(288 + 23t + -8t2)' equal to zero and attempt to solve: Simplifying 288 + 23t + -8t2 = 0 Solving 288 + 23t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -36 + -2.875t + t2 = 0 Move the constant term to the right: Add '36' to each side of the equation. -36 + -2.875t + 36 + t2 = 0 + 36 Reorder the terms: -36 + 36 + -2.875t + t2 = 0 + 36 Combine like terms: -36 + 36 = 0 0 + -2.875t + t2 = 0 + 36 -2.875t + t2 = 0 + 36 Combine like terms: 0 + 36 = 36 -2.875t + t2 = 36 The t term is -2.875t. Take half its coefficient (-1.4375). Square it (2.06640625) and add it to both sides. Add '2.06640625' to each side of the equation. -2.875t + 2.06640625 + t2 = 36 + 2.06640625 Reorder the terms: 2.06640625 + -2.875t + t2 = 36 + 2.06640625 Combine like terms: 36 + 2.06640625 = 38.06640625 2.06640625 + -2.875t + t2 = 38.06640625 Factor a perfect square on the left side: (t + -1.4375)(t + -1.4375) = 38.06640625 Calculate the square root of the right side: 6.16979791 Break this problem into two subproblems by setting (t + -1.4375) equal to 6.16979791 and -6.16979791.Subproblem 1
t + -1.4375 = 6.16979791 Simplifying t + -1.4375 = 6.16979791 Reorder the terms: -1.4375 + t = 6.16979791 Solving -1.4375 + t = 6.16979791 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.4375' to each side of the equation. -1.4375 + 1.4375 + t = 6.16979791 + 1.4375 Combine like terms: -1.4375 + 1.4375 = 0.0000 0.0000 + t = 6.16979791 + 1.4375 t = 6.16979791 + 1.4375 Combine like terms: 6.16979791 + 1.4375 = 7.60729791 t = 7.60729791 Simplifying t = 7.60729791Subproblem 2
t + -1.4375 = -6.16979791 Simplifying t + -1.4375 = -6.16979791 Reorder the terms: -1.4375 + t = -6.16979791 Solving -1.4375 + t = -6.16979791 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.4375' to each side of the equation. -1.4375 + 1.4375 + t = -6.16979791 + 1.4375 Combine like terms: -1.4375 + 1.4375 = 0.0000 0.0000 + t = -6.16979791 + 1.4375 t = -6.16979791 + 1.4375 Combine like terms: -6.16979791 + 1.4375 = -4.73229791 t = -4.73229791 Simplifying t = -4.73229791Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.60729791, -4.73229791}Solution
t = {7.60729791, -4.73229791}
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