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Simplifying 32 = -4.9t2 + 14.5t + 25 Reorder the terms: 32 = 25 + 14.5t + -4.9t2 Solving 32 = 25 + 14.5t + -4.9t2 Solving for variable 't'. Combine like terms: 32 + -25 = 7 7 + -14.5t + 4.9t2 = 25 + 14.5t + -4.9t2 + -25 + -14.5t + 4.9t2 Reorder the terms: 7 + -14.5t + 4.9t2 = 25 + -25 + 14.5t + -14.5t + -4.9t2 + 4.9t2 Combine like terms: 25 + -25 = 0 7 + -14.5t + 4.9t2 = 0 + 14.5t + -14.5t + -4.9t2 + 4.9t2 7 + -14.5t + 4.9t2 = 14.5t + -14.5t + -4.9t2 + 4.9t2 Combine like terms: 14.5t + -14.5t = 0.0 7 + -14.5t + 4.9t2 = 0.0 + -4.9t2 + 4.9t2 7 + -14.5t + 4.9t2 = -4.9t2 + 4.9t2 Combine like terms: -4.9t2 + 4.9t2 = 0.0 7 + -14.5t + 4.9t2 = 0.0 Begin completing the square. Divide all terms by 4.9 the coefficient of the squared term: Divide each side by '4.9'. 1.428571429 + -2.959183673t + t2 = 0 Move the constant term to the right: Add '-1.428571429' to each side of the equation. 1.428571429 + -2.959183673t + -1.428571429 + t2 = 0 + -1.428571429 Reorder the terms: 1.428571429 + -1.428571429 + -2.959183673t + t2 = 0 + -1.428571429 Combine like terms: 1.428571429 + -1.428571429 = 0.000000000 0.000000000 + -2.959183673t + t2 = 0 + -1.428571429 -2.959183673t + t2 = 0 + -1.428571429 Combine like terms: 0 + -1.428571429 = -1.428571429 -2.959183673t + t2 = -1.428571429 The t term is -2.959183673t. Take half its coefficient (-1.479591837). Square it (2.189192004) and add it to both sides. Add '2.189192004' to each side of the equation. -2.959183673t + 2.189192004 + t2 = -1.428571429 + 2.189192004 Reorder the terms: 2.189192004 + -2.959183673t + t2 = -1.428571429 + 2.189192004 Combine like terms: -1.428571429 + 2.189192004 = 0.760620575 2.189192004 + -2.959183673t + t2 = 0.760620575 Factor a perfect square on the left side: (t + -1.479591837)(t + -1.479591837) = 0.760620575 Calculate the square root of the right side: 0.87213564 Break this problem into two subproblems by setting (t + -1.479591837) equal to 0.87213564 and -0.87213564.Subproblem 1
t + -1.479591837 = 0.87213564 Simplifying t + -1.479591837 = 0.87213564 Reorder the terms: -1.479591837 + t = 0.87213564 Solving -1.479591837 + t = 0.87213564 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.479591837' to each side of the equation. -1.479591837 + 1.479591837 + t = 0.87213564 + 1.479591837 Combine like terms: -1.479591837 + 1.479591837 = 0.000000000 0.000000000 + t = 0.87213564 + 1.479591837 t = 0.87213564 + 1.479591837 Combine like terms: 0.87213564 + 1.479591837 = 2.351727477 t = 2.351727477 Simplifying t = 2.351727477Subproblem 2
t + -1.479591837 = -0.87213564 Simplifying t + -1.479591837 = -0.87213564 Reorder the terms: -1.479591837 + t = -0.87213564 Solving -1.479591837 + t = -0.87213564 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.479591837' to each side of the equation. -1.479591837 + 1.479591837 + t = -0.87213564 + 1.479591837 Combine like terms: -1.479591837 + 1.479591837 = 0.000000000 0.000000000 + t = -0.87213564 + 1.479591837 t = -0.87213564 + 1.479591837 Combine like terms: -0.87213564 + 1.479591837 = 0.607456197 t = 0.607456197 Simplifying t = 0.607456197Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.351727477, 0.607456197}
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