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Simplifying -16t2 + 12t + 6 = 0 Reorder the terms: 6 + 12t + -16t2 = 0 Solving 6 + 12t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(3 + 6t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(3 + 6t + -8t2)' equal to zero and attempt to solve: Simplifying 3 + 6t + -8t2 = 0 Solving 3 + 6t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.375 + -0.75t + t2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + -0.75t + 0.375 + t2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + -0.75t + t2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + -0.75t + t2 = 0 + 0.375 -0.75t + t2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 -0.75t + t2 = 0.375 The t term is -0.75t. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75t + 0.140625 + t2 = 0.375 + 0.140625 Reorder the terms: 0.140625 + -0.75t + t2 = 0.375 + 0.140625 Combine like terms: 0.375 + 0.140625 = 0.515625 0.140625 + -0.75t + t2 = 0.515625 Factor a perfect square on the left side: (t + -0.375)(t + -0.375) = 0.515625 Calculate the square root of the right side: 0.718070331 Break this problem into two subproblems by setting (t + -0.375) equal to 0.718070331 and -0.718070331.Subproblem 1
t + -0.375 = 0.718070331 Simplifying t + -0.375 = 0.718070331 Reorder the terms: -0.375 + t = 0.718070331 Solving -0.375 + t = 0.718070331 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + t = 0.718070331 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = 0.718070331 + 0.375 t = 0.718070331 + 0.375 Combine like terms: 0.718070331 + 0.375 = 1.093070331 t = 1.093070331 Simplifying t = 1.093070331Subproblem 2
t + -0.375 = -0.718070331 Simplifying t + -0.375 = -0.718070331 Reorder the terms: -0.375 + t = -0.718070331 Solving -0.375 + t = -0.718070331 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + t = -0.718070331 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = -0.718070331 + 0.375 t = -0.718070331 + 0.375 Combine like terms: -0.718070331 + 0.375 = -0.343070331 t = -0.343070331 Simplifying t = -0.343070331Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.093070331, -0.343070331}Solution
t = {1.093070331, -0.343070331}
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