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Simplifying -16t2 + 96t + 32 = 0 Reorder the terms: 32 + 96t + -16t2 = 0 Solving 32 + 96t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '16'. 16(2 + 6t + -1t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(2 + 6t + -1t2)' equal to zero and attempt to solve: Simplifying 2 + 6t + -1t2 = 0 Solving 2 + 6t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -2 + -6t + t2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -6t + 2 + t2 = 0 + 2 Reorder the terms: -2 + 2 + -6t + t2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -6t + t2 = 0 + 2 -6t + t2 = 0 + 2 Combine like terms: 0 + 2 = 2 -6t + t2 = 2 The t term is -6t. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6t + 9 + t2 = 2 + 9 Reorder the terms: 9 + -6t + t2 = 2 + 9 Combine like terms: 2 + 9 = 11 9 + -6t + t2 = 11 Factor a perfect square on the left side: (t + -3)(t + -3) = 11 Calculate the square root of the right side: 3.31662479 Break this problem into two subproblems by setting (t + -3) equal to 3.31662479 and -3.31662479.Subproblem 1
t + -3 = 3.31662479 Simplifying t + -3 = 3.31662479 Reorder the terms: -3 + t = 3.31662479 Solving -3 + t = 3.31662479 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = 3.31662479 + 3 Combine like terms: -3 + 3 = 0 0 + t = 3.31662479 + 3 t = 3.31662479 + 3 Combine like terms: 3.31662479 + 3 = 6.31662479 t = 6.31662479 Simplifying t = 6.31662479Subproblem 2
t + -3 = -3.31662479 Simplifying t + -3 = -3.31662479 Reorder the terms: -3 + t = -3.31662479 Solving -3 + t = -3.31662479 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = -3.31662479 + 3 Combine like terms: -3 + 3 = 0 0 + t = -3.31662479 + 3 t = -3.31662479 + 3 Combine like terms: -3.31662479 + 3 = -0.31662479 t = -0.31662479 Simplifying t = -0.31662479Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.31662479, -0.31662479}Solution
t = {6.31662479, -0.31662479}
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