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Simplifying 16t2 + 8t + -192 = 0 Reorder the terms: -192 + 8t + 16t2 = 0 Solving -192 + 8t + 16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '8'. 8(-24 + t + 2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-24 + t + 2t2)' equal to zero and attempt to solve: Simplifying -24 + t + 2t2 = 0 Solving -24 + t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -12 + 0.5t + t2 = 0 Move the constant term to the right: Add '12' to each side of the equation. -12 + 0.5t + 12 + t2 = 0 + 12 Reorder the terms: -12 + 12 + 0.5t + t2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + 0.5t + t2 = 0 + 12 0.5t + t2 = 0 + 12 Combine like terms: 0 + 12 = 12 0.5t + t2 = 12 The t term is t. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.5t + 0.25 + t2 = 12 + 0.25 Reorder the terms: 0.25 + 0.5t + t2 = 12 + 0.25 Combine like terms: 12 + 0.25 = 12.25 0.25 + 0.5t + t2 = 12.25 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 12.25 Calculate the square root of the right side: 3.5 Break this problem into two subproblems by setting (t + 0.5) equal to 3.5 and -3.5.Subproblem 1
t + 0.5 = 3.5 Simplifying t + 0.5 = 3.5 Reorder the terms: 0.5 + t = 3.5 Solving 0.5 + t = 3.5 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = 3.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 3.5 + -0.5 t = 3.5 + -0.5 Combine like terms: 3.5 + -0.5 = 3 t = 3 Simplifying t = 3Subproblem 2
t + 0.5 = -3.5 Simplifying t + 0.5 = -3.5 Reorder the terms: 0.5 + t = -3.5 Solving 0.5 + t = -3.5 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = -3.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -3.5 + -0.5 t = -3.5 + -0.5 Combine like terms: -3.5 + -0.5 = -4 t = -4 Simplifying t = -4Solution
The solution to the problem is based on the solutions from the subproblems. t = {3, -4}Solution
t = {3, -4}
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