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Simplifying 20 = 4 + 36t + -16t2 Solving 20 = 4 + 36t + -16t2 Solving for variable 't'. Combine like terms: 20 + -4 = 16 16 + -36t + 16t2 = 4 + 36t + -16t2 + -4 + -36t + 16t2 Reorder the terms: 16 + -36t + 16t2 = 4 + -4 + 36t + -36t + -16t2 + 16t2 Combine like terms: 4 + -4 = 0 16 + -36t + 16t2 = 0 + 36t + -36t + -16t2 + 16t2 16 + -36t + 16t2 = 36t + -36t + -16t2 + 16t2 Combine like terms: 36t + -36t = 0 16 + -36t + 16t2 = 0 + -16t2 + 16t2 16 + -36t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 16 + -36t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(4 + -9t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(4 + -9t + 4t2)' equal to zero and attempt to solve: Simplifying 4 + -9t + 4t2 = 0 Solving 4 + -9t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 1 + -2.25t + t2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -2.25t + -1 + t2 = 0 + -1 Reorder the terms: 1 + -1 + -2.25t + t2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -2.25t + t2 = 0 + -1 -2.25t + t2 = 0 + -1 Combine like terms: 0 + -1 = -1 -2.25t + t2 = -1 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 = -1 + 1.265625 Reorder the terms: 1.265625 + -2.25t + t2 = -1 + 1.265625 Combine like terms: -1 + 1.265625 = 0.265625 1.265625 + -2.25t + t2 = 0.265625 Factor a perfect square on the left side: (t + -1.125)(t + -1.125) = 0.265625 Calculate the square root of the right side: 0.515388203 Break this problem into two subproblems by setting (t + -1.125) equal to 0.515388203 and -0.515388203.Subproblem 1
t + -1.125 = 0.515388203 Simplifying t + -1.125 = 0.515388203 Reorder the terms: -1.125 + t = 0.515388203 Solving -1.125 + t = 0.515388203 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 = 0.515388203 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + t = 0.515388203 + 1.125 t = 0.515388203 + 1.125 Combine like terms: 0.515388203 + 1.125 = 1.640388203 t = 1.640388203 Simplifying t = 1.640388203Subproblem 2
t + -1.125 = -0.515388203 Simplifying t + -1.125 = -0.515388203 Reorder the terms: -1.125 + t = -0.515388203 Solving -1.125 + t = -0.515388203 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 = -0.515388203 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + t = -0.515388203 + 1.125 t = -0.515388203 + 1.125 Combine like terms: -0.515388203 + 1.125 = 0.609611797 t = 0.609611797 Simplifying t = 0.609611797Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.640388203, 0.609611797}Solution
t = {1.640388203, 0.609611797}
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