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Simplifying 11 = 3 + 28t + -16t2 Solving 11 = 3 + 28t + -16t2 Solving for variable 't'. Combine like terms: 11 + -3 = 8 8 + -28t + 16t2 = 3 + 28t + -16t2 + -3 + -28t + 16t2 Reorder the terms: 8 + -28t + 16t2 = 3 + -3 + 28t + -28t + -16t2 + 16t2 Combine like terms: 3 + -3 = 0 8 + -28t + 16t2 = 0 + 28t + -28t + -16t2 + 16t2 8 + -28t + 16t2 = 28t + -28t + -16t2 + 16t2 Combine like terms: 28t + -28t = 0 8 + -28t + 16t2 = 0 + -16t2 + 16t2 8 + -28t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 8 + -28t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(2 + -7t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(2 + -7t + 4t2)' equal to zero and attempt to solve: Simplifying 2 + -7t + 4t2 = 0 Solving 2 + -7t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.5 + -1.75t + t2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -1.75t + -0.5 + t2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -1.75t + t2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -1.75t + t2 = 0 + -0.5 -1.75t + t2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -1.75t + t2 = -0.5 The t term is -1.75t. Take half its coefficient (-0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. -1.75t + 0.765625 + t2 = -0.5 + 0.765625 Reorder the terms: 0.765625 + -1.75t + t2 = -0.5 + 0.765625 Combine like terms: -0.5 + 0.765625 = 0.265625 0.765625 + -1.75t + t2 = 0.265625 Factor a perfect square on the left side: (t + -0.875)(t + -0.875) = 0.265625 Calculate the square root of the right side: 0.515388203 Break this problem into two subproblems by setting (t + -0.875) equal to 0.515388203 and -0.515388203.Subproblem 1
t + -0.875 = 0.515388203 Simplifying t + -0.875 = 0.515388203 Reorder the terms: -0.875 + t = 0.515388203 Solving -0.875 + t = 0.515388203 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + t = 0.515388203 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + t = 0.515388203 + 0.875 t = 0.515388203 + 0.875 Combine like terms: 0.515388203 + 0.875 = 1.390388203 t = 1.390388203 Simplifying t = 1.390388203Subproblem 2
t + -0.875 = -0.515388203 Simplifying t + -0.875 = -0.515388203 Reorder the terms: -0.875 + t = -0.515388203 Solving -0.875 + t = -0.515388203 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + t = -0.515388203 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + t = -0.515388203 + 0.875 t = -0.515388203 + 0.875 Combine like terms: -0.515388203 + 0.875 = 0.359611797 t = 0.359611797 Simplifying t = 0.359611797Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.390388203, 0.359611797}Solution
t = {1.390388203, 0.359611797}
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