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Simplifying 180 = 16t2 + 28t + 24 Reorder the terms: 180 = 24 + 28t + 16t2 Solving 180 = 24 + 28t + 16t2 Solving for variable 't'. Combine like terms: 180 + -24 = 156 156 + -28t + -16t2 = 24 + 28t + 16t2 + -24 + -28t + -16t2 Reorder the terms: 156 + -28t + -16t2 = 24 + -24 + 28t + -28t + 16t2 + -16t2 Combine like terms: 24 + -24 = 0 156 + -28t + -16t2 = 0 + 28t + -28t + 16t2 + -16t2 156 + -28t + -16t2 = 28t + -28t + 16t2 + -16t2 Combine like terms: 28t + -28t = 0 156 + -28t + -16t2 = 0 + 16t2 + -16t2 156 + -28t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 156 + -28t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(39 + -7t + -4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(39 + -7t + -4t2)' equal to zero and attempt to solve: Simplifying 39 + -7t + -4t2 = 0 Solving 39 + -7t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -9.75 + 1.75t + t2 = 0 Move the constant term to the right: Add '9.75' to each side of the equation. -9.75 + 1.75t + 9.75 + t2 = 0 + 9.75 Reorder the terms: -9.75 + 9.75 + 1.75t + t2 = 0 + 9.75 Combine like terms: -9.75 + 9.75 = 0.00 0.00 + 1.75t + t2 = 0 + 9.75 1.75t + t2 = 0 + 9.75 Combine like terms: 0 + 9.75 = 9.75 1.75t + t2 = 9.75 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 = 9.75 + 0.765625 Reorder the terms: 0.765625 + 1.75t + t2 = 9.75 + 0.765625 Combine like terms: 9.75 + 0.765625 = 10.515625 0.765625 + 1.75t + t2 = 10.515625 Factor a perfect square on the left side: (t + 0.875)(t + 0.875) = 10.515625 Calculate the square root of the right side: 3.242780443 Break this problem into two subproblems by setting (t + 0.875) equal to 3.242780443 and -3.242780443.Subproblem 1
t + 0.875 = 3.242780443 Simplifying t + 0.875 = 3.242780443 Reorder the terms: 0.875 + t = 3.242780443 Solving 0.875 + t = 3.242780443 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 = 3.242780443 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + t = 3.242780443 + -0.875 t = 3.242780443 + -0.875 Combine like terms: 3.242780443 + -0.875 = 2.367780443 t = 2.367780443 Simplifying t = 2.367780443Subproblem 2
t + 0.875 = -3.242780443 Simplifying t + 0.875 = -3.242780443 Reorder the terms: 0.875 + t = -3.242780443 Solving 0.875 + t = -3.242780443 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 = -3.242780443 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + t = -3.242780443 + -0.875 t = -3.242780443 + -0.875 Combine like terms: -3.242780443 + -0.875 = -4.117780443 t = -4.117780443 Simplifying t = -4.117780443Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.367780443, -4.117780443}Solution
t = {2.367780443, -4.117780443}
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