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Simplifying -16t2 + 32t + 14 = 0 Reorder the terms: 14 + 32t + -16t2 = 0 Solving 14 + 32t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(7 + 16t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(7 + 16t + -8t2)' equal to zero and attempt to solve: Simplifying 7 + 16t + -8t2 = 0 Solving 7 + 16t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.875 + -2t + t2 = 0 Move the constant term to the right: Add '0.875' to each side of the equation. -0.875 + -2t + 0.875 + t2 = 0 + 0.875 Reorder the terms: -0.875 + 0.875 + -2t + t2 = 0 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + -2t + t2 = 0 + 0.875 -2t + t2 = 0 + 0.875 Combine like terms: 0 + 0.875 = 0.875 -2t + t2 = 0.875 The t term is -2t. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2t + 1 + t2 = 0.875 + 1 Reorder the terms: 1 + -2t + t2 = 0.875 + 1 Combine like terms: 0.875 + 1 = 1.875 1 + -2t + t2 = 1.875 Factor a perfect square on the left side: (t + -1)(t + -1) = 1.875 Calculate the square root of the right side: 1.369306394 Break this problem into two subproblems by setting (t + -1) equal to 1.369306394 and -1.369306394.Subproblem 1
t + -1 = 1.369306394 Simplifying t + -1 = 1.369306394 Reorder the terms: -1 + t = 1.369306394 Solving -1 + t = 1.369306394 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = 1.369306394 + 1 Combine like terms: -1 + 1 = 0 0 + t = 1.369306394 + 1 t = 1.369306394 + 1 Combine like terms: 1.369306394 + 1 = 2.369306394 t = 2.369306394 Simplifying t = 2.369306394Subproblem 2
t + -1 = -1.369306394 Simplifying t + -1 = -1.369306394 Reorder the terms: -1 + t = -1.369306394 Solving -1 + t = -1.369306394 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = -1.369306394 + 1 Combine like terms: -1 + 1 = 0 0 + t = -1.369306394 + 1 t = -1.369306394 + 1 Combine like terms: -1.369306394 + 1 = -0.369306394 t = -0.369306394 Simplifying t = -0.369306394Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.369306394, -0.369306394}Solution
t = {2.369306394, -0.369306394}
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