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Simplifying 32 = -16t2 + 64t + -4 Reorder the terms: 32 = -4 + 64t + -16t2 Solving 32 = -4 + 64t + -16t2 Solving for variable 't'. Combine like terms: 32 + 4 = 36 36 + -64t + 16t2 = -4 + 64t + -16t2 + 4 + -64t + 16t2 Reorder the terms: 36 + -64t + 16t2 = -4 + 4 + 64t + -64t + -16t2 + 16t2 Combine like terms: -4 + 4 = 0 36 + -64t + 16t2 = 0 + 64t + -64t + -16t2 + 16t2 36 + -64t + 16t2 = 64t + -64t + -16t2 + 16t2 Combine like terms: 64t + -64t = 0 36 + -64t + 16t2 = 0 + -16t2 + 16t2 36 + -64t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 36 + -64t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(9 + -16t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(9 + -16t + 4t2)' equal to zero and attempt to solve: Simplifying 9 + -16t + 4t2 = 0 Solving 9 + -16t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 2.25 + -4t + t2 = 0 Move the constant term to the right: Add '-2.25' to each side of the equation. 2.25 + -4t + -2.25 + t2 = 0 + -2.25 Reorder the terms: 2.25 + -2.25 + -4t + t2 = 0 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + -4t + t2 = 0 + -2.25 -4t + t2 = 0 + -2.25 Combine like terms: 0 + -2.25 = -2.25 -4t + t2 = -2.25 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 = -2.25 + 4 Reorder the terms: 4 + -4t + t2 = -2.25 + 4 Combine like terms: -2.25 + 4 = 1.75 4 + -4t + t2 = 1.75 Factor a perfect square on the left side: (t + -2)(t + -2) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (t + -2) equal to 1.322875656 and -1.322875656.Subproblem 1
t + -2 = 1.322875656 Simplifying t + -2 = 1.322875656 Reorder the terms: -2 + t = 1.322875656 Solving -2 + t = 1.322875656 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 = 1.322875656 + 2 Combine like terms: -2 + 2 = 0 0 + t = 1.322875656 + 2 t = 1.322875656 + 2 Combine like terms: 1.322875656 + 2 = 3.322875656 t = 3.322875656 Simplifying t = 3.322875656Subproblem 2
t + -2 = -1.322875656 Simplifying t + -2 = -1.322875656 Reorder the terms: -2 + t = -1.322875656 Solving -2 + t = -1.322875656 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 = -1.322875656 + 2 Combine like terms: -2 + 2 = 0 0 + t = -1.322875656 + 2 t = -1.322875656 + 2 Combine like terms: -1.322875656 + 2 = 0.677124344 t = 0.677124344 Simplifying t = 0.677124344Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.322875656, 0.677124344}Solution
t = {3.322875656, 0.677124344}
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