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Simplifying 8(t + 4) * t = 6 Reorder the terms: 8(4 + t) * t = 6 Reorder the terms for easier multiplication: 8t(4 + t) = 6 (4 * 8t + t * 8t) = 6 (32t + 8t2) = 6 Solving 32t + 8t2 = 6 Solving for variable 't'. Reorder the terms: -6 + 32t + 8t2 = 6 + -6 Combine like terms: 6 + -6 = 0 -6 + 32t + 8t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + 16t + 4t2) = 0 Ignore the factor 2.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 = 4.75 4 + 4t + t2 = 4.75 Factor a perfect square on the left side: (t + 2)(t + 2) = 4.75 Calculate the square root of the right side: 2.179449472 Break this problem into two subproblems by setting (t + 2) equal to 2.179449472 and -2.179449472.Subproblem 1
t + 2 = 2.179449472 Simplifying t + 2 = 2.179449472 Reorder the terms: 2 + t = 2.179449472 Solving 2 + t = 2.179449472 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 = 2.179449472 + -2 Combine like terms: 2 + -2 = 0 0 + t = 2.179449472 + -2 t = 2.179449472 + -2 Combine like terms: 2.179449472 + -2 = 0.179449472 t = 0.179449472 Simplifying t = 0.179449472Subproblem 2
t + 2 = -2.179449472 Simplifying t + 2 = -2.179449472 Reorder the terms: 2 + t = -2.179449472 Solving 2 + t = -2.179449472 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 = -2.179449472 + -2 Combine like terms: 2 + -2 = 0 0 + t = -2.179449472 + -2 t = -2.179449472 + -2 Combine like terms: -2.179449472 + -2 = -4.179449472 t = -4.179449472 Simplifying t = -4.179449472Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.179449472, -4.179449472}Solution
t = {0.179449472, -4.179449472}
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