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Simplifying 20 = -16t2 + 110t + 80 Reorder the terms: 20 = 80 + 110t + -16t2 Solving 20 = 80 + 110t + -16t2 Solving for variable 't'. Combine like terms: 20 + -80 = -60 -60 + -110t + 16t2 = 80 + 110t + -16t2 + -80 + -110t + 16t2 Reorder the terms: -60 + -110t + 16t2 = 80 + -80 + 110t + -110t + -16t2 + 16t2 Combine like terms: 80 + -80 = 0 -60 + -110t + 16t2 = 0 + 110t + -110t + -16t2 + 16t2 -60 + -110t + 16t2 = 110t + -110t + -16t2 + 16t2 Combine like terms: 110t + -110t = 0 -60 + -110t + 16t2 = 0 + -16t2 + 16t2 -60 + -110t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -60 + -110t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-30 + -55t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-30 + -55t + 8t2)' equal to zero and attempt to solve: Simplifying -30 + -55t + 8t2 = 0 Solving -30 + -55t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -3.75 + -6.875t + t2 = 0 Move the constant term to the right: Add '3.75' to each side of the equation. -3.75 + -6.875t + 3.75 + t2 = 0 + 3.75 Reorder the terms: -3.75 + 3.75 + -6.875t + t2 = 0 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + -6.875t + t2 = 0 + 3.75 -6.875t + t2 = 0 + 3.75 Combine like terms: 0 + 3.75 = 3.75 -6.875t + t2 = 3.75 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 = 3.75 + 11.81640625 Reorder the terms: 11.81640625 + -6.875t + t2 = 3.75 + 11.81640625 Combine like terms: 3.75 + 11.81640625 = 15.56640625 11.81640625 + -6.875t + t2 = 15.56640625 Factor a perfect square on the left side: (t + -3.4375)(t + -3.4375) = 15.56640625 Calculate the square root of the right side: 3.945428526 Break this problem into two subproblems by setting (t + -3.4375) equal to 3.945428526 and -3.945428526.Subproblem 1
t + -3.4375 = 3.945428526 Simplifying t + -3.4375 = 3.945428526 Reorder the terms: -3.4375 + t = 3.945428526 Solving -3.4375 + t = 3.945428526 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 = 3.945428526 + 3.4375 Combine like terms: -3.4375 + 3.4375 = 0.0000 0.0000 + t = 3.945428526 + 3.4375 t = 3.945428526 + 3.4375 Combine like terms: 3.945428526 + 3.4375 = 7.382928526 t = 7.382928526 Simplifying t = 7.382928526Subproblem 2
t + -3.4375 = -3.945428526 Simplifying t + -3.4375 = -3.945428526 Reorder the terms: -3.4375 + t = -3.945428526 Solving -3.4375 + t = -3.945428526 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 = -3.945428526 + 3.4375 Combine like terms: -3.4375 + 3.4375 = 0.0000 0.0000 + t = -3.945428526 + 3.4375 t = -3.945428526 + 3.4375 Combine like terms: -3.945428526 + 3.4375 = -0.507928526 t = -0.507928526 Simplifying t = -0.507928526Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.382928526, -0.507928526}Solution
t = {7.382928526, -0.507928526}
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