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Simplifying 0 = 280 + 110t + -16t2 Solving 0 = 280 + 110t + -16t2 Solving for variable 't'. Combine like terms: 0 + -280 = -280 -280 + -110t + 16t2 = 280 + 110t + -16t2 + -280 + -110t + 16t2 Reorder the terms: -280 + -110t + 16t2 = 280 + -280 + 110t + -110t + -16t2 + 16t2 Combine like terms: 280 + -280 = 0 -280 + -110t + 16t2 = 0 + 110t + -110t + -16t2 + 16t2 -280 + -110t + 16t2 = 110t + -110t + -16t2 + 16t2 Combine like terms: 110t + -110t = 0 -280 + -110t + 16t2 = 0 + -16t2 + 16t2 -280 + -110t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -280 + -110t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-140 + -55t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-140 + -55t + 8t2)' equal to zero and attempt to solve: Simplifying -140 + -55t + 8t2 = 0 Solving -140 + -55t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -17.5 + -6.875t + t2 = 0 Move the constant term to the right: Add '17.5' to each side of the equation. -17.5 + -6.875t + 17.5 + t2 = 0 + 17.5 Reorder the terms: -17.5 + 17.5 + -6.875t + t2 = 0 + 17.5 Combine like terms: -17.5 + 17.5 = 0.0 0.0 + -6.875t + t2 = 0 + 17.5 -6.875t + t2 = 0 + 17.5 Combine like terms: 0 + 17.5 = 17.5 -6.875t + t2 = 17.5 The t term is -6.875t. Take half its coefficient (-3.4375). Square it (11.81640625) and add it to both sides. Add '11.81640625' to each side of the equation. -6.875t + 11.81640625 + t2 = 17.5 + 11.81640625 Reorder the terms: 11.81640625 + -6.875t + t2 = 17.5 + 11.81640625 Combine like terms: 17.5 + 11.81640625 = 29.31640625 11.81640625 + -6.875t + t2 = 29.31640625 Factor a perfect square on the left side: (t + -3.4375)(t + -3.4375) = 29.31640625 Calculate the square root of the right side: 5.414462693 Break this problem into two subproblems by setting (t + -3.4375) equal to 5.414462693 and -5.414462693.Subproblem 1
t + -3.4375 = 5.414462693 Simplifying t + -3.4375 = 5.414462693 Reorder the terms: -3.4375 + t = 5.414462693 Solving -3.4375 + t = 5.414462693 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.4375' to each side of the equation. -3.4375 + 3.4375 + t = 5.414462693 + 3.4375 Combine like terms: -3.4375 + 3.4375 = 0.0000 0.0000 + t = 5.414462693 + 3.4375 t = 5.414462693 + 3.4375 Combine like terms: 5.414462693 + 3.4375 = 8.851962693 t = 8.851962693 Simplifying t = 8.851962693Subproblem 2
t + -3.4375 = -5.414462693 Simplifying t + -3.4375 = -5.414462693 Reorder the terms: -3.4375 + t = -5.414462693 Solving -3.4375 + t = -5.414462693 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.4375' to each side of the equation. -3.4375 + 3.4375 + t = -5.414462693 + 3.4375 Combine like terms: -3.4375 + 3.4375 = 0.0000 0.0000 + t = -5.414462693 + 3.4375 t = -5.414462693 + 3.4375 Combine like terms: -5.414462693 + 3.4375 = -1.976962693 t = -1.976962693 Simplifying t = -1.976962693Solution
The solution to the problem is based on the solutions from the subproblems. t = {8.851962693, -1.976962693}Solution
t = {8.851962693, -1.976962693}
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