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Simplifying 0 = 4.9t * t + 20t + 12 Multiply t * t 0 = 4.9t2 + 20t + 12 Reorder the terms: 0 = 12 + 20t + 4.9t2 Solving 0 = 12 + 20t + 4.9t2 Solving for variable 't'. Combine like terms: 0 + -12 = -12 -12 + -20t + -4.9t2 = 12 + 20t + 4.9t2 + -12 + -20t + -4.9t2 Reorder the terms: -12 + -20t + -4.9t2 = 12 + -12 + 20t + -20t + 4.9t2 + -4.9t2 Combine like terms: 12 + -12 = 0 -12 + -20t + -4.9t2 = 0 + 20t + -20t + 4.9t2 + -4.9t2 -12 + -20t + -4.9t2 = 20t + -20t + 4.9t2 + -4.9t2 Combine like terms: 20t + -20t = 0 -12 + -20t + -4.9t2 = 0 + 4.9t2 + -4.9t2 -12 + -20t + -4.9t2 = 4.9t2 + -4.9t2 Combine like terms: 4.9t2 + -4.9t2 = 0.0 -12 + -20t + -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'. 2.448979592 + 4.081632653t + t2 = 0 Move the constant term to the right: Add '-2.448979592' to each side of the equation. 2.448979592 + 4.081632653t + -2.448979592 + t2 = 0 + -2.448979592 Reorder the terms: 2.448979592 + -2.448979592 + 4.081632653t + t2 = 0 + -2.448979592 Combine like terms: 2.448979592 + -2.448979592 = 0.000000000 0.000000000 + 4.081632653t + t2 = 0 + -2.448979592 4.081632653t + t2 = 0 + -2.448979592 Combine like terms: 0 + -2.448979592 = -2.448979592 4.081632653t + t2 = -2.448979592 The t term is 4.081632653t. Take half its coefficient (2.040816327). Square it (4.164931281) and add it to both sides. Add '4.164931281' to each side of the equation. 4.081632653t + 4.164931281 + t2 = -2.448979592 + 4.164931281 Reorder the terms: 4.164931281 + 4.081632653t + t2 = -2.448979592 + 4.164931281 Combine like terms: -2.448979592 + 4.164931281 = 1.715951689 4.164931281 + 4.081632653t + t2 = 1.715951689 Factor a perfect square on the left side: (t + 2.040816327)(t + 2.040816327) = 1.715951689 Calculate the square root of the right side: 1.309943392 Break this problem into two subproblems by setting (t + 2.040816327) equal to 1.309943392 and -1.309943392.Subproblem 1
t + 2.040816327 = 1.309943392 Simplifying t + 2.040816327 = 1.309943392 Reorder the terms: 2.040816327 + t = 1.309943392 Solving 2.040816327 + t = 1.309943392 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.040816327' to each side of the equation. 2.040816327 + -2.040816327 + t = 1.309943392 + -2.040816327 Combine like terms: 2.040816327 + -2.040816327 = 0.000000000 0.000000000 + t = 1.309943392 + -2.040816327 t = 1.309943392 + -2.040816327 Combine like terms: 1.309943392 + -2.040816327 = -0.730872935 t = -0.730872935 Simplifying t = -0.730872935Subproblem 2
t + 2.040816327 = -1.309943392 Simplifying t + 2.040816327 = -1.309943392 Reorder the terms: 2.040816327 + t = -1.309943392 Solving 2.040816327 + t = -1.309943392 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.040816327' to each side of the equation. 2.040816327 + -2.040816327 + t = -1.309943392 + -2.040816327 Combine like terms: 2.040816327 + -2.040816327 = 0.000000000 0.000000000 + t = -1.309943392 + -2.040816327 t = -1.309943392 + -2.040816327 Combine like terms: -1.309943392 + -2.040816327 = -3.350759719 t = -3.350759719 Simplifying t = -3.350759719Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.730872935, -3.350759719}
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