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Simplifying 3(5 + -1t) * 4t = 18 Reorder the terms for easier multiplication: 3 * 4t(5 + -1t) = 18 Multiply 3 * 4 12t(5 + -1t) = 18 (5 * 12t + -1t * 12t) = 18 (60t + -12t2) = 18 Solving 60t + -12t2 = 18 Solving for variable 't'. Reorder the terms: -18 + 60t + -12t2 = 18 + -18 Combine like terms: 18 + -18 = 0 -18 + 60t + -12t2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-3 + 10t + -2t2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-3 + 10t + -2t2)' equal to zero and attempt to solve: Simplifying -3 + 10t + -2t2 = 0 Solving -3 + 10t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 1.5 + -5t + t2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -5t + -1.5 + t2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -5t + t2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -5t + t2 = 0 + -1.5 -5t + t2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -5t + t2 = -1.5 The t term is -5t. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5t + 6.25 + t2 = -1.5 + 6.25 Reorder the terms: 6.25 + -5t + t2 = -1.5 + 6.25 Combine like terms: -1.5 + 6.25 = 4.75 6.25 + -5t + t2 = 4.75 Factor a perfect square on the left side: (t + -2.5)(t + -2.5) = 4.75 Calculate the square root of the right side: 2.179449472 Break this problem into two subproblems by setting (t + -2.5) equal to 2.179449472 and -2.179449472.Subproblem 1
t + -2.5 = 2.179449472 Simplifying t + -2.5 = 2.179449472 Reorder the terms: -2.5 + t = 2.179449472 Solving -2.5 + t = 2.179449472 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = 2.179449472 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = 2.179449472 + 2.5 t = 2.179449472 + 2.5 Combine like terms: 2.179449472 + 2.5 = 4.679449472 t = 4.679449472 Simplifying t = 4.679449472Subproblem 2
t + -2.5 = -2.179449472 Simplifying t + -2.5 = -2.179449472 Reorder the terms: -2.5 + t = -2.179449472 Solving -2.5 + t = -2.179449472 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = -2.179449472 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = -2.179449472 + 2.5 t = -2.179449472 + 2.5 Combine like terms: -2.179449472 + 2.5 = 0.320550528 t = 0.320550528 Simplifying t = 0.320550528Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.679449472, 0.320550528}Solution
t = {4.679449472, 0.320550528}
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