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Simplifying 256 = 16t2 + 64t + 192 Reorder the terms: 256 = 192 + 64t + 16t2 Solving 256 = 192 + 64t + 16t2 Solving for variable 't'. Combine like terms: 256 + -192 = 64 64 + -64t + -16t2 = 192 + 64t + 16t2 + -192 + -64t + -16t2 Reorder the terms: 64 + -64t + -16t2 = 192 + -192 + 64t + -64t + 16t2 + -16t2 Combine like terms: 192 + -192 = 0 64 + -64t + -16t2 = 0 + 64t + -64t + 16t2 + -16t2 64 + -64t + -16t2 = 64t + -64t + 16t2 + -16t2 Combine like terms: 64t + -64t = 0 64 + -64t + -16t2 = 0 + 16t2 + -16t2 64 + -64t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 64 + -64t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '16'. 16(4 + -4t + -1t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(4 + -4t + -1t2)' equal to zero and attempt to solve: Simplifying 4 + -4t + -1t2 = 0 Solving 4 + -4t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -4 + 4t + t2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 4t + 4 + t2 = 0 + 4 Reorder the terms: -4 + 4 + 4t + t2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 4t + t2 = 0 + 4 4t + t2 = 0 + 4 Combine like terms: 0 + 4 = 4 4t + t2 = 4 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 = 4 + 4 Reorder the terms: 4 + 4t + t2 = 4 + 4 Combine like terms: 4 + 4 = 8 4 + 4t + t2 = 8 Factor a perfect square on the left side: (t + 2)(t + 2) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (t + 2) equal to 2.828427125 and -2.828427125.Subproblem 1
t + 2 = 2.828427125 Simplifying t + 2 = 2.828427125 Reorder the terms: 2 + t = 2.828427125 Solving 2 + t = 2.828427125 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 = 2.828427125 + -2 Combine like terms: 2 + -2 = 0 0 + t = 2.828427125 + -2 t = 2.828427125 + -2 Combine like terms: 2.828427125 + -2 = 0.828427125 t = 0.828427125 Simplifying t = 0.828427125Subproblem 2
t + 2 = -2.828427125 Simplifying t + 2 = -2.828427125 Reorder the terms: 2 + t = -2.828427125 Solving 2 + t = -2.828427125 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 = -2.828427125 + -2 Combine like terms: 2 + -2 = 0 0 + t = -2.828427125 + -2 t = -2.828427125 + -2 Combine like terms: -2.828427125 + -2 = -4.828427125 t = -4.828427125 Simplifying t = -4.828427125Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.828427125, -4.828427125}Solution
t = {0.828427125, -4.828427125}
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