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Simplifying 0 = 16t2 + 48t + -32 Reorder the terms: 0 = -32 + 48t + 16t2 Solving 0 = -32 + 48t + 16t2 Solving for variable 't'. Combine like terms: 0 + 32 = 32 32 + -48t + -16t2 = -32 + 48t + 16t2 + 32 + -48t + -16t2 Reorder the terms: 32 + -48t + -16t2 = -32 + 32 + 48t + -48t + 16t2 + -16t2 Combine like terms: -32 + 32 = 0 32 + -48t + -16t2 = 0 + 48t + -48t + 16t2 + -16t2 32 + -48t + -16t2 = 48t + -48t + 16t2 + -16t2 Combine like terms: 48t + -48t = 0 32 + -48t + -16t2 = 0 + 16t2 + -16t2 32 + -48t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 32 + -48t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '16'. 16(2 + -3t + -1t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(2 + -3t + -1t2)' equal to zero and attempt to solve: Simplifying 2 + -3t + -1t2 = 0 Solving 2 + -3t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -2 + 3t + t2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 3t + 2 + t2 = 0 + 2 Reorder the terms: -2 + 2 + 3t + t2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 3t + t2 = 0 + 2 3t + t2 = 0 + 2 Combine like terms: 0 + 2 = 2 3t + t2 = 2 The t term is 3t. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3t + 2.25 + t2 = 2 + 2.25 Reorder the terms: 2.25 + 3t + t2 = 2 + 2.25 Combine like terms: 2 + 2.25 = 4.25 2.25 + 3t + t2 = 4.25 Factor a perfect square on the left side: (t + 1.5)(t + 1.5) = 4.25 Calculate the square root of the right side: 2.061552813 Break this problem into two subproblems by setting (t + 1.5) equal to 2.061552813 and -2.061552813.Subproblem 1
t + 1.5 = 2.061552813 Simplifying t + 1.5 = 2.061552813 Reorder the terms: 1.5 + t = 2.061552813 Solving 1.5 + t = 2.061552813 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + t = 2.061552813 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = 2.061552813 + -1.5 t = 2.061552813 + -1.5 Combine like terms: 2.061552813 + -1.5 = 0.561552813 t = 0.561552813 Simplifying t = 0.561552813Subproblem 2
t + 1.5 = -2.061552813 Simplifying t + 1.5 = -2.061552813 Reorder the terms: 1.5 + t = -2.061552813 Solving 1.5 + t = -2.061552813 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + t = -2.061552813 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = -2.061552813 + -1.5 t = -2.061552813 + -1.5 Combine like terms: -2.061552813 + -1.5 = -3.561552813 t = -3.561552813 Simplifying t = -3.561552813Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.561552813, -3.561552813}Solution
t = {0.561552813, -3.561552813}
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