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Simplifying 20 = -16t2 + 36t + 100 Reorder the terms: 20 = 100 + 36t + -16t2 Solving 20 = 100 + 36t + -16t2 Solving for variable 't'. Combine like terms: 20 + -100 = -80 -80 + -36t + 16t2 = 100 + 36t + -16t2 + -100 + -36t + 16t2 Reorder the terms: -80 + -36t + 16t2 = 100 + -100 + 36t + -36t + -16t2 + 16t2 Combine like terms: 100 + -100 = 0 -80 + -36t + 16t2 = 0 + 36t + -36t + -16t2 + 16t2 -80 + -36t + 16t2 = 36t + -36t + -16t2 + 16t2 Combine like terms: 36t + -36t = 0 -80 + -36t + 16t2 = 0 + -16t2 + 16t2 -80 + -36t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -80 + -36t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-20 + -9t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-20 + -9t + 4t2)' equal to zero and attempt to solve: Simplifying -20 + -9t + 4t2 = 0 Solving -20 + -9t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -5 + -2.25t + t2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -2.25t + 5 + t2 = 0 + 5 Reorder the terms: -5 + 5 + -2.25t + t2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -2.25t + t2 = 0 + 5 -2.25t + t2 = 0 + 5 Combine like terms: 0 + 5 = 5 -2.25t + t2 = 5 The t term is -2.25t. Take half its coefficient (-1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. -2.25t + 1.265625 + t2 = 5 + 1.265625 Reorder the terms: 1.265625 + -2.25t + t2 = 5 + 1.265625 Combine like terms: 5 + 1.265625 = 6.265625 1.265625 + -2.25t + t2 = 6.265625 Factor a perfect square on the left side: (t + -1.125)(t + -1.125) = 6.265625 Calculate the square root of the right side: 2.503123049 Break this problem into two subproblems by setting (t + -1.125) equal to 2.503123049 and -2.503123049.Subproblem 1
t + -1.125 = 2.503123049 Simplifying t + -1.125 = 2.503123049 Reorder the terms: -1.125 + t = 2.503123049 Solving -1.125 + t = 2.503123049 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + t = 2.503123049 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + t = 2.503123049 + 1.125 t = 2.503123049 + 1.125 Combine like terms: 2.503123049 + 1.125 = 3.628123049 t = 3.628123049 Simplifying t = 3.628123049Subproblem 2
t + -1.125 = -2.503123049 Simplifying t + -1.125 = -2.503123049 Reorder the terms: -1.125 + t = -2.503123049 Solving -1.125 + t = -2.503123049 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + t = -2.503123049 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + t = -2.503123049 + 1.125 t = -2.503123049 + 1.125 Combine like terms: -2.503123049 + 1.125 = -1.378123049 t = -1.378123049 Simplifying t = -1.378123049Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.628123049, -1.378123049}Solution
t = {3.628123049, -1.378123049}
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