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Simplifying 6t(4t + -1) = 5 Reorder the terms: 6t(-1 + 4t) = 5 (-1 * 6t + 4t * 6t) = 5 (-6t + 24t2) = 5 Solving -6t + 24t2 = 5 Solving for variable 't'. Reorder the terms: -5 + -6t + 24t2 = 5 + -5 Combine like terms: 5 + -5 = 0 -5 + -6t + 24t2 = 0 Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -0.2083333333 + -0.25t + t2 = 0 Move the constant term to the right: Add '0.2083333333' to each side of the equation. -0.2083333333 + -0.25t + 0.2083333333 + t2 = 0 + 0.2083333333 Reorder the terms: -0.2083333333 + 0.2083333333 + -0.25t + t2 = 0 + 0.2083333333 Combine like terms: -0.2083333333 + 0.2083333333 = 0.0000000000 0.0000000000 + -0.25t + t2 = 0 + 0.2083333333 -0.25t + t2 = 0 + 0.2083333333 Combine like terms: 0 + 0.2083333333 = 0.2083333333 -0.25t + t2 = 0.2083333333 The t term is -0.25t. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25t + 0.015625 + t2 = 0.2083333333 + 0.015625 Reorder the terms: 0.015625 + -0.25t + t2 = 0.2083333333 + 0.015625 Combine like terms: 0.2083333333 + 0.015625 = 0.2239583333 0.015625 + -0.25t + t2 = 0.2239583333 Factor a perfect square on the left side: (t + -0.125)(t + -0.125) = 0.2239583333 Calculate the square root of the right side: 0.473242362 Break this problem into two subproblems by setting (t + -0.125) equal to 0.473242362 and -0.473242362.Subproblem 1
t + -0.125 = 0.473242362 Simplifying t + -0.125 = 0.473242362 Reorder the terms: -0.125 + t = 0.473242362 Solving -0.125 + t = 0.473242362 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + t = 0.473242362 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + t = 0.473242362 + 0.125 t = 0.473242362 + 0.125 Combine like terms: 0.473242362 + 0.125 = 0.598242362 t = 0.598242362 Simplifying t = 0.598242362Subproblem 2
t + -0.125 = -0.473242362 Simplifying t + -0.125 = -0.473242362 Reorder the terms: -0.125 + t = -0.473242362 Solving -0.125 + t = -0.473242362 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + t = -0.473242362 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + t = -0.473242362 + 0.125 t = -0.473242362 + 0.125 Combine like terms: -0.473242362 + 0.125 = -0.348242362 t = -0.348242362 Simplifying t = -0.348242362Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.598242362, -0.348242362}
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