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Simplifying 4t3 + -64t2 + -128t = 0 Reorder the terms: -128t + -64t2 + 4t3 = 0 Solving -128t + -64t2 + 4t3 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '4t'. 4t(-32 + -16t + t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor 't' equal to zero and attempt to solve: Simplifying t = 0 Solving t = 0 Move all terms containing t to the left, all other terms to the right. Simplifying t = 0Subproblem 2
Set the factor '(-32 + -16t + t2)' equal to zero and attempt to solve: Simplifying -32 + -16t + t2 = 0 Solving -32 + -16t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '32' to each side of the equation. -32 + -16t + 32 + t2 = 0 + 32 Reorder the terms: -32 + 32 + -16t + t2 = 0 + 32 Combine like terms: -32 + 32 = 0 0 + -16t + t2 = 0 + 32 -16t + t2 = 0 + 32 Combine like terms: 0 + 32 = 32 -16t + t2 = 32 The t term is -16t. Take half its coefficient (-8). Square it (64) and add it to both sides. Add '64' to each side of the equation. -16t + 64 + t2 = 32 + 64 Reorder the terms: 64 + -16t + t2 = 32 + 64 Combine like terms: 32 + 64 = 96 64 + -16t + t2 = 96 Factor a perfect square on the left side: (t + -8)(t + -8) = 96 Calculate the square root of the right side: 9.797958971 Break this problem into two subproblems by setting (t + -8) equal to 9.797958971 and -9.797958971.Subproblem 1
t + -8 = 9.797958971 Simplifying t + -8 = 9.797958971 Reorder the terms: -8 + t = 9.797958971 Solving -8 + t = 9.797958971 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + t = 9.797958971 + 8 Combine like terms: -8 + 8 = 0 0 + t = 9.797958971 + 8 t = 9.797958971 + 8 Combine like terms: 9.797958971 + 8 = 17.797958971 t = 17.797958971 Simplifying t = 17.797958971Subproblem 2
t + -8 = -9.797958971 Simplifying t + -8 = -9.797958971 Reorder the terms: -8 + t = -9.797958971 Solving -8 + t = -9.797958971 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + t = -9.797958971 + 8 Combine like terms: -8 + 8 = 0 0 + t = -9.797958971 + 8 t = -9.797958971 + 8 Combine like terms: -9.797958971 + 8 = -1.797958971 t = -1.797958971 Simplifying t = -1.797958971Solution
The solution to the problem is based on the solutions from the subproblems. t = {17.797958971, -1.797958971}Solution
t = {0, 17.797958971, -1.797958971}
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