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Simplifying 0 = 16t2 + 64t + 12 Reorder the terms: 0 = 12 + 64t + 16t2 Solving 0 = 12 + 64t + 16t2 Solving for variable 't'. Combine like terms: 0 + -12 = -12 -12 + -64t + -16t2 = 12 + 64t + 16t2 + -12 + -64t + -16t2 Reorder the terms: -12 + -64t + -16t2 = 12 + -12 + 64t + -64t + 16t2 + -16t2 Combine like terms: 12 + -12 = 0 -12 + -64t + -16t2 = 0 + 64t + -64t + 16t2 + -16t2 -12 + -64t + -16t2 = 64t + -64t + 16t2 + -16t2 Combine like terms: 64t + -64t = 0 -12 + -64t + -16t2 = 0 + 16t2 + -16t2 -12 + -64t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -12 + -64t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-4'. -4(3 + 16t + 4t2) = 0 Ignore the factor -4.Subproblem 1
Set the factor '(3 + 16t + 4t2)' equal to zero and attempt to solve: Simplifying 3 + 16t + 4t2 = 0 Solving 3 + 16t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.75 + 4t + t2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + 4t + -0.75 + t2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + 4t + t2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + 4t + t2 = 0 + -0.75 4t + t2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 4t + t2 = -0.75 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 = -0.75 + 4 Reorder the terms: 4 + 4t + t2 = -0.75 + 4 Combine like terms: -0.75 + 4 = 3.25 4 + 4t + t2 = 3.25 Factor a perfect square on the left side: (t + 2)(t + 2) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (t + 2) equal to 1.802775638 and -1.802775638.Subproblem 1
t + 2 = 1.802775638 Simplifying t + 2 = 1.802775638 Reorder the terms: 2 + t = 1.802775638 Solving 2 + t = 1.802775638 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 = 1.802775638 + -2 Combine like terms: 2 + -2 = 0 0 + t = 1.802775638 + -2 t = 1.802775638 + -2 Combine like terms: 1.802775638 + -2 = -0.197224362 t = -0.197224362 Simplifying t = -0.197224362Subproblem 2
t + 2 = -1.802775638 Simplifying t + 2 = -1.802775638 Reorder the terms: 2 + t = -1.802775638 Solving 2 + t = -1.802775638 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 = -1.802775638 + -2 Combine like terms: 2 + -2 = 0 0 + t = -1.802775638 + -2 t = -1.802775638 + -2 Combine like terms: -1.802775638 + -2 = -3.802775638 t = -3.802775638 Simplifying t = -3.802775638Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.197224362, -3.802775638}Solution
t = {-0.197224362, -3.802775638}
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