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Simplifying 48 = -16t2 + 48t + 196 Reorder the terms: 48 = 196 + 48t + -16t2 Solving 48 = 196 + 48t + -16t2 Solving for variable 't'. Combine like terms: 48 + -196 = -148 -148 + -48t + 16t2 = 196 + 48t + -16t2 + -196 + -48t + 16t2 Reorder the terms: -148 + -48t + 16t2 = 196 + -196 + 48t + -48t + -16t2 + 16t2 Combine like terms: 196 + -196 = 0 -148 + -48t + 16t2 = 0 + 48t + -48t + -16t2 + 16t2 -148 + -48t + 16t2 = 48t + -48t + -16t2 + 16t2 Combine like terms: 48t + -48t = 0 -148 + -48t + 16t2 = 0 + -16t2 + 16t2 -148 + -48t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -148 + -48t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-37 + -12t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-37 + -12t + 4t2)' equal to zero and attempt to solve: Simplifying -37 + -12t + 4t2 = 0 Solving -37 + -12t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -9.25 + -3t + t2 = 0 Move the constant term to the right: Add '9.25' to each side of the equation. -9.25 + -3t + 9.25 + t2 = 0 + 9.25 Reorder the terms: -9.25 + 9.25 + -3t + t2 = 0 + 9.25 Combine like terms: -9.25 + 9.25 = 0.00 0.00 + -3t + t2 = 0 + 9.25 -3t + t2 = 0 + 9.25 Combine like terms: 0 + 9.25 = 9.25 -3t + t2 = 9.25 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 = 9.25 + 2.25 Reorder the terms: 2.25 + -3t + t2 = 9.25 + 2.25 Combine like terms: 9.25 + 2.25 = 11.5 2.25 + -3t + t2 = 11.5 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 11.5 Calculate the square root of the right side: 3.391164992 Break this problem into two subproblems by setting (t + -1.5) equal to 3.391164992 and -3.391164992.Subproblem 1
t + -1.5 = 3.391164992 Simplifying t + -1.5 = 3.391164992 Reorder the terms: -1.5 + t = 3.391164992 Solving -1.5 + t = 3.391164992 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 = 3.391164992 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 3.391164992 + 1.5 t = 3.391164992 + 1.5 Combine like terms: 3.391164992 + 1.5 = 4.891164992 t = 4.891164992 Simplifying t = 4.891164992Subproblem 2
t + -1.5 = -3.391164992 Simplifying t + -1.5 = -3.391164992 Reorder the terms: -1.5 + t = -3.391164992 Solving -1.5 + t = -3.391164992 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 = -3.391164992 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -3.391164992 + 1.5 t = -3.391164992 + 1.5 Combine like terms: -3.391164992 + 1.5 = -1.891164992 t = -1.891164992 Simplifying t = -1.891164992Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.891164992, -1.891164992}Solution
t = {4.891164992, -1.891164992}
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