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Simplifying 0 = -16t2 + 90t + 160 Reorder the terms: 0 = 160 + 90t + -16t2 Solving 0 = 160 + 90t + -16t2 Solving for variable 't'. Combine like terms: 0 + -160 = -160 -160 + -90t + 16t2 = 160 + 90t + -16t2 + -160 + -90t + 16t2 Reorder the terms: -160 + -90t + 16t2 = 160 + -160 + 90t + -90t + -16t2 + 16t2 Combine like terms: 160 + -160 = 0 -160 + -90t + 16t2 = 0 + 90t + -90t + -16t2 + 16t2 -160 + -90t + 16t2 = 90t + -90t + -16t2 + 16t2 Combine like terms: 90t + -90t = 0 -160 + -90t + 16t2 = 0 + -16t2 + 16t2 -160 + -90t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -160 + -90t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-80 + -45t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-80 + -45t + 8t2)' equal to zero and attempt to solve: Simplifying -80 + -45t + 8t2 = 0 Solving -80 + -45t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -10 + -5.625t + t2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + -5.625t + 10 + t2 = 0 + 10 Reorder the terms: -10 + 10 + -5.625t + t2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + -5.625t + t2 = 0 + 10 -5.625t + t2 = 0 + 10 Combine like terms: 0 + 10 = 10 -5.625t + t2 = 10 The t term is -5.625t. Take half its coefficient (-2.8125). Square it (7.91015625) and add it to both sides. Add '7.91015625' to each side of the equation. -5.625t + 7.91015625 + t2 = 10 + 7.91015625 Reorder the terms: 7.91015625 + -5.625t + t2 = 10 + 7.91015625 Combine like terms: 10 + 7.91015625 = 17.91015625 7.91015625 + -5.625t + t2 = 17.91015625 Factor a perfect square on the left side: (t + -2.8125)(t + -2.8125) = 17.91015625 Calculate the square root of the right side: 4.232039254 Break this problem into two subproblems by setting (t + -2.8125) equal to 4.232039254 and -4.232039254.Subproblem 1
t + -2.8125 = 4.232039254 Simplifying t + -2.8125 = 4.232039254 Reorder the terms: -2.8125 + t = 4.232039254 Solving -2.8125 + t = 4.232039254 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.8125' to each side of the equation. -2.8125 + 2.8125 + t = 4.232039254 + 2.8125 Combine like terms: -2.8125 + 2.8125 = 0.0000 0.0000 + t = 4.232039254 + 2.8125 t = 4.232039254 + 2.8125 Combine like terms: 4.232039254 + 2.8125 = 7.044539254 t = 7.044539254 Simplifying t = 7.044539254Subproblem 2
t + -2.8125 = -4.232039254 Simplifying t + -2.8125 = -4.232039254 Reorder the terms: -2.8125 + t = -4.232039254 Solving -2.8125 + t = -4.232039254 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.8125' to each side of the equation. -2.8125 + 2.8125 + t = -4.232039254 + 2.8125 Combine like terms: -2.8125 + 2.8125 = 0.0000 0.0000 + t = -4.232039254 + 2.8125 t = -4.232039254 + 2.8125 Combine like terms: -4.232039254 + 2.8125 = -1.419539254 t = -1.419539254 Simplifying t = -1.419539254Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.044539254, -1.419539254}Solution
t = {7.044539254, -1.419539254}
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