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Simplifying -16t2 + 6t + 12 = 0 Reorder the terms: 12 + 6t + -16t2 = 0 Solving 12 + 6t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(6 + 3t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(6 + 3t + -8t2)' equal to zero and attempt to solve: Simplifying 6 + 3t + -8t2 = 0 Solving 6 + 3t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.75 + -0.375t + t2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -0.375t + 0.75 + t2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -0.375t + t2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -0.375t + t2 = 0 + 0.75 -0.375t + t2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -0.375t + t2 = 0.75 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 = 0.75 + 0.03515625 Reorder the terms: 0.03515625 + -0.375t + t2 = 0.75 + 0.03515625 Combine like terms: 0.75 + 0.03515625 = 0.78515625 0.03515625 + -0.375t + t2 = 0.78515625 Factor a perfect square on the left side: (t + -0.1875)(t + -0.1875) = 0.78515625 Calculate the square root of the right side: 0.88609043 Break this problem into two subproblems by setting (t + -0.1875) equal to 0.88609043 and -0.88609043.Subproblem 1
t + -0.1875 = 0.88609043 Simplifying t + -0.1875 = 0.88609043 Reorder the terms: -0.1875 + t = 0.88609043 Solving -0.1875 + t = 0.88609043 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 = 0.88609043 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + t = 0.88609043 + 0.1875 t = 0.88609043 + 0.1875 Combine like terms: 0.88609043 + 0.1875 = 1.07359043 t = 1.07359043 Simplifying t = 1.07359043Subproblem 2
t + -0.1875 = -0.88609043 Simplifying t + -0.1875 = -0.88609043 Reorder the terms: -0.1875 + t = -0.88609043 Solving -0.1875 + t = -0.88609043 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 = -0.88609043 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + t = -0.88609043 + 0.1875 t = -0.88609043 + 0.1875 Combine like terms: -0.88609043 + 0.1875 = -0.69859043 t = -0.69859043 Simplifying t = -0.69859043Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.07359043, -0.69859043}Solution
t = {1.07359043, -0.69859043}
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