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Simplifying 0 = -6t2 + 24t + 14 Reorder the terms: 0 = 14 + 24t + -6t2 Solving 0 = 14 + 24t + -6t2 Solving for variable 't'. Combine like terms: 0 + -14 = -14 -14 + -24t + 6t2 = 14 + 24t + -6t2 + -14 + -24t + 6t2 Reorder the terms: -14 + -24t + 6t2 = 14 + -14 + 24t + -24t + -6t2 + 6t2 Combine like terms: 14 + -14 = 0 -14 + -24t + 6t2 = 0 + 24t + -24t + -6t2 + 6t2 -14 + -24t + 6t2 = 24t + -24t + -6t2 + 6t2 Combine like terms: 24t + -24t = 0 -14 + -24t + 6t2 = 0 + -6t2 + 6t2 -14 + -24t + 6t2 = -6t2 + 6t2 Combine like terms: -6t2 + 6t2 = 0 -14 + -24t + 6t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + -12t + 3t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + -12t + 3t2)' equal to zero and attempt to solve: Simplifying -7 + -12t + 3t2 = 0 Solving -7 + -12t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2.333333333 + -4t + t2 = 0 Move the constant term to the right: Add '2.333333333' to each side of the equation. -2.333333333 + -4t + 2.333333333 + t2 = 0 + 2.333333333 Reorder the terms: -2.333333333 + 2.333333333 + -4t + t2 = 0 + 2.333333333 Combine like terms: -2.333333333 + 2.333333333 = 0.000000000 0.000000000 + -4t + t2 = 0 + 2.333333333 -4t + t2 = 0 + 2.333333333 Combine like terms: 0 + 2.333333333 = 2.333333333 -4t + t2 = 2.333333333 The t term is -4t. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4t + 4 + t2 = 2.333333333 + 4 Reorder the terms: 4 + -4t + t2 = 2.333333333 + 4 Combine like terms: 2.333333333 + 4 = 6.333333333 4 + -4t + t2 = 6.333333333 Factor a perfect square on the left side: (t + -2)(t + -2) = 6.333333333 Calculate the square root of the right side: 2.516611478 Break this problem into two subproblems by setting (t + -2) equal to 2.516611478 and -2.516611478.Subproblem 1
t + -2 = 2.516611478 Simplifying t + -2 = 2.516611478 Reorder the terms: -2 + t = 2.516611478 Solving -2 + t = 2.516611478 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = 2.516611478 + 2 Combine like terms: -2 + 2 = 0 0 + t = 2.516611478 + 2 t = 2.516611478 + 2 Combine like terms: 2.516611478 + 2 = 4.516611478 t = 4.516611478 Simplifying t = 4.516611478Subproblem 2
t + -2 = -2.516611478 Simplifying t + -2 = -2.516611478 Reorder the terms: -2 + t = -2.516611478 Solving -2 + t = -2.516611478 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = -2.516611478 + 2 Combine like terms: -2 + 2 = 0 0 + t = -2.516611478 + 2 t = -2.516611478 + 2 Combine like terms: -2.516611478 + 2 = -0.516611478 t = -0.516611478 Simplifying t = -0.516611478Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.516611478, -0.516611478}Solution
t = {4.516611478, -0.516611478}
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