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Simplifying -14t2 + 80t + 20 = 0 Reorder the terms: 20 + 80t + -14t2 = 0 Solving 20 + 80t + -14t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(10 + 40t + -7t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(10 + 40t + -7t2)' equal to zero and attempt to solve: Simplifying 10 + 40t + -7t2 = 0 Solving 10 + 40t + -7t2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -1.428571429 + -5.714285714t + t2 = 0 Move the constant term to the right: Add '1.428571429' to each side of the equation. -1.428571429 + -5.714285714t + 1.428571429 + t2 = 0 + 1.428571429 Reorder the terms: -1.428571429 + 1.428571429 + -5.714285714t + t2 = 0 + 1.428571429 Combine like terms: -1.428571429 + 1.428571429 = 0.000000000 0.000000000 + -5.714285714t + t2 = 0 + 1.428571429 -5.714285714t + t2 = 0 + 1.428571429 Combine like terms: 0 + 1.428571429 = 1.428571429 -5.714285714t + t2 = 1.428571429 The t term is -5.714285714t. Take half its coefficient (-2.857142857). Square it (8.163265305) and add it to both sides. Add '8.163265305' to each side of the equation. -5.714285714t + 8.163265305 + t2 = 1.428571429 + 8.163265305 Reorder the terms: 8.163265305 + -5.714285714t + t2 = 1.428571429 + 8.163265305 Combine like terms: 1.428571429 + 8.163265305 = 9.591836734 8.163265305 + -5.714285714t + t2 = 9.591836734 Factor a perfect square on the left side: (t + -2.857142857)(t + -2.857142857) = 9.591836734 Calculate the square root of the right side: 3.097069055 Break this problem into two subproblems by setting (t + -2.857142857) equal to 3.097069055 and -3.097069055.Subproblem 1
t + -2.857142857 = 3.097069055 Simplifying t + -2.857142857 = 3.097069055 Reorder the terms: -2.857142857 + t = 3.097069055 Solving -2.857142857 + t = 3.097069055 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.857142857' to each side of the equation. -2.857142857 + 2.857142857 + t = 3.097069055 + 2.857142857 Combine like terms: -2.857142857 + 2.857142857 = 0.000000000 0.000000000 + t = 3.097069055 + 2.857142857 t = 3.097069055 + 2.857142857 Combine like terms: 3.097069055 + 2.857142857 = 5.954211912 t = 5.954211912 Simplifying t = 5.954211912Subproblem 2
t + -2.857142857 = -3.097069055 Simplifying t + -2.857142857 = -3.097069055 Reorder the terms: -2.857142857 + t = -3.097069055 Solving -2.857142857 + t = -3.097069055 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.857142857' to each side of the equation. -2.857142857 + 2.857142857 + t = -3.097069055 + 2.857142857 Combine like terms: -2.857142857 + 2.857142857 = 0.000000000 0.000000000 + t = -3.097069055 + 2.857142857 t = -3.097069055 + 2.857142857 Combine like terms: -3.097069055 + 2.857142857 = -0.239926198 t = -0.239926198 Simplifying t = -0.239926198Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.954211912, -0.239926198}Solution
t = {5.954211912, -0.239926198}
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