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Simplifying 3 = 16t2 + -19t Reorder the terms: 3 = -19t + 16t2 Solving 3 = -19t + 16t2 Solving for variable 't'. Reorder the terms: 3 + 19t + -16t2 = -19t + 19t + 16t2 + -16t2 Combine like terms: -19t + 19t = 0 3 + 19t + -16t2 = 0 + 16t2 + -16t2 3 + 19t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 3 + 19t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.1875 + -1.1875t + t2 = 0 Move the constant term to the right: Add '0.1875' to each side of the equation. -0.1875 + -1.1875t + 0.1875 + t2 = 0 + 0.1875 Reorder the terms: -0.1875 + 0.1875 + -1.1875t + t2 = 0 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + -1.1875t + t2 = 0 + 0.1875 -1.1875t + t2 = 0 + 0.1875 Combine like terms: 0 + 0.1875 = 0.1875 -1.1875t + t2 = 0.1875 The t term is -1.1875t. Take half its coefficient (-0.59375). Square it (0.3525390625) and add it to both sides. Add '0.3525390625' to each side of the equation. -1.1875t + 0.3525390625 + t2 = 0.1875 + 0.3525390625 Reorder the terms: 0.3525390625 + -1.1875t + t2 = 0.1875 + 0.3525390625 Combine like terms: 0.1875 + 0.3525390625 = 0.5400390625 0.3525390625 + -1.1875t + t2 = 0.5400390625 Factor a perfect square on the left side: (t + -0.59375)(t + -0.59375) = 0.5400390625 Calculate the square root of the right side: 0.734873501 Break this problem into two subproblems by setting (t + -0.59375) equal to 0.734873501 and -0.734873501.Subproblem 1
t + -0.59375 = 0.734873501 Simplifying t + -0.59375 = 0.734873501 Reorder the terms: -0.59375 + t = 0.734873501 Solving -0.59375 + t = 0.734873501 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.59375' to each side of the equation. -0.59375 + 0.59375 + t = 0.734873501 + 0.59375 Combine like terms: -0.59375 + 0.59375 = 0.00000 0.00000 + t = 0.734873501 + 0.59375 t = 0.734873501 + 0.59375 Combine like terms: 0.734873501 + 0.59375 = 1.328623501 t = 1.328623501 Simplifying t = 1.328623501Subproblem 2
t + -0.59375 = -0.734873501 Simplifying t + -0.59375 = -0.734873501 Reorder the terms: -0.59375 + t = -0.734873501 Solving -0.59375 + t = -0.734873501 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.59375' to each side of the equation. -0.59375 + 0.59375 + t = -0.734873501 + 0.59375 Combine like terms: -0.59375 + 0.59375 = 0.00000 0.00000 + t = -0.734873501 + 0.59375 t = -0.734873501 + 0.59375 Combine like terms: -0.734873501 + 0.59375 = -0.141123501 t = -0.141123501 Simplifying t = -0.141123501Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.328623501, -0.141123501}
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