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Simplifying 108 = 16t2 + 64t + 80 Reorder the terms: 108 = 80 + 64t + 16t2 Solving 108 = 80 + 64t + 16t2 Solving for variable 't'. Combine like terms: 108 + -80 = 28 28 + -64t + -16t2 = 80 + 64t + 16t2 + -80 + -64t + -16t2 Reorder the terms: 28 + -64t + -16t2 = 80 + -80 + 64t + -64t + 16t2 + -16t2 Combine like terms: 80 + -80 = 0 28 + -64t + -16t2 = 0 + 64t + -64t + 16t2 + -16t2 28 + -64t + -16t2 = 64t + -64t + 16t2 + -16t2 Combine like terms: 64t + -64t = 0 28 + -64t + -16t2 = 0 + 16t2 + -16t2 28 + -64t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 28 + -64t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(7 + -16t + -4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(7 + -16t + -4t2)' equal to zero and attempt to solve: Simplifying 7 + -16t + -4t2 = 0 Solving 7 + -16t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -1.75 + 4t + t2 = 0 Move the constant term to the right: Add '1.75' to each side of the equation. -1.75 + 4t + 1.75 + t2 = 0 + 1.75 Reorder the terms: -1.75 + 1.75 + 4t + t2 = 0 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + 4t + t2 = 0 + 1.75 4t + t2 = 0 + 1.75 Combine like terms: 0 + 1.75 = 1.75 4t + t2 = 1.75 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 = 1.75 + 4 Reorder the terms: 4 + 4t + t2 = 1.75 + 4 Combine like terms: 1.75 + 4 = 5.75 4 + 4t + t2 = 5.75 Factor a perfect square on the left side: (t + 2)(t + 2) = 5.75 Calculate the square root of the right side: 2.397915762 Break this problem into two subproblems by setting (t + 2) equal to 2.397915762 and -2.397915762.Subproblem 1
t + 2 = 2.397915762 Simplifying t + 2 = 2.397915762 Reorder the terms: 2 + t = 2.397915762 Solving 2 + t = 2.397915762 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 = 2.397915762 + -2 Combine like terms: 2 + -2 = 0 0 + t = 2.397915762 + -2 t = 2.397915762 + -2 Combine like terms: 2.397915762 + -2 = 0.397915762 t = 0.397915762 Simplifying t = 0.397915762Subproblem 2
t + 2 = -2.397915762 Simplifying t + 2 = -2.397915762 Reorder the terms: 2 + t = -2.397915762 Solving 2 + t = -2.397915762 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 = -2.397915762 + -2 Combine like terms: 2 + -2 = 0 0 + t = -2.397915762 + -2 t = -2.397915762 + -2 Combine like terms: -2.397915762 + -2 = -4.397915762 t = -4.397915762 Simplifying t = -4.397915762Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.397915762, -4.397915762}Solution
t = {0.397915762, -4.397915762}
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