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Simplifying -16t2 + 6t + 31 = 0 Reorder the terms: 31 + 6t + -16t2 = 0 Solving 31 + 6t + -16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -1.9375 + -0.375t + t2 = 0 Move the constant term to the right: Add '1.9375' to each side of the equation. -1.9375 + -0.375t + 1.9375 + t2 = 0 + 1.9375 Reorder the terms: -1.9375 + 1.9375 + -0.375t + t2 = 0 + 1.9375 Combine like terms: -1.9375 + 1.9375 = 0.0000 0.0000 + -0.375t + t2 = 0 + 1.9375 -0.375t + t2 = 0 + 1.9375 Combine like terms: 0 + 1.9375 = 1.9375 -0.375t + t2 = 1.9375 The t term is -0.375t. Take half its coefficient (-0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. -0.375t + 0.03515625 + t2 = 1.9375 + 0.03515625 Reorder the terms: 0.03515625 + -0.375t + t2 = 1.9375 + 0.03515625 Combine like terms: 1.9375 + 0.03515625 = 1.97265625 0.03515625 + -0.375t + t2 = 1.97265625 Factor a perfect square on the left side: (t + -0.1875)(t + -0.1875) = 1.97265625 Calculate the square root of the right side: 1.404512816 Break this problem into two subproblems by setting (t + -0.1875) equal to 1.404512816 and -1.404512816.Subproblem 1
t + -0.1875 = 1.404512816 Simplifying t + -0.1875 = 1.404512816 Reorder the terms: -0.1875 + t = 1.404512816 Solving -0.1875 + t = 1.404512816 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + t = 1.404512816 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + t = 1.404512816 + 0.1875 t = 1.404512816 + 0.1875 Combine like terms: 1.404512816 + 0.1875 = 1.592012816 t = 1.592012816 Simplifying t = 1.592012816Subproblem 2
t + -0.1875 = -1.404512816 Simplifying t + -0.1875 = -1.404512816 Reorder the terms: -0.1875 + t = -1.404512816 Solving -0.1875 + t = -1.404512816 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + t = -1.404512816 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + t = -1.404512816 + 0.1875 t = -1.404512816 + 0.1875 Combine like terms: -1.404512816 + 0.1875 = -1.217012816 t = -1.217012816 Simplifying t = -1.217012816Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.592012816, -1.217012816}
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