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Simplifying 24t2 + 30t + 5 = 0 Reorder the terms: 5 + 30t + 24t2 = 0 Solving 5 + 30t + 24t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. 0.2083333333 + 1.25t + t2 = 0 Move the constant term to the right: Add '-0.2083333333' to each side of the equation. 0.2083333333 + 1.25t + -0.2083333333 + t2 = 0 + -0.2083333333 Reorder the terms: 0.2083333333 + -0.2083333333 + 1.25t + t2 = 0 + -0.2083333333 Combine like terms: 0.2083333333 + -0.2083333333 = 0.0000000000 0.0000000000 + 1.25t + t2 = 0 + -0.2083333333 1.25t + t2 = 0 + -0.2083333333 Combine like terms: 0 + -0.2083333333 = -0.2083333333 1.25t + t2 = -0.2083333333 The t term is 1.25t. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25t + 0.390625 + t2 = -0.2083333333 + 0.390625 Reorder the terms: 0.390625 + 1.25t + t2 = -0.2083333333 + 0.390625 Combine like terms: -0.2083333333 + 0.390625 = 0.1822916667 0.390625 + 1.25t + t2 = 0.1822916667 Factor a perfect square on the left side: (t + 0.625)(t + 0.625) = 0.1822916667 Calculate the square root of the right side: 0.426956282 Break this problem into two subproblems by setting (t + 0.625) equal to 0.426956282 and -0.426956282.Subproblem 1
t + 0.625 = 0.426956282 Simplifying t + 0.625 = 0.426956282 Reorder the terms: 0.625 + t = 0.426956282 Solving 0.625 + t = 0.426956282 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + t = 0.426956282 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + t = 0.426956282 + -0.625 t = 0.426956282 + -0.625 Combine like terms: 0.426956282 + -0.625 = -0.198043718 t = -0.198043718 Simplifying t = -0.198043718Subproblem 2
t + 0.625 = -0.426956282 Simplifying t + 0.625 = -0.426956282 Reorder the terms: 0.625 + t = -0.426956282 Solving 0.625 + t = -0.426956282 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + t = -0.426956282 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + t = -0.426956282 + -0.625 t = -0.426956282 + -0.625 Combine like terms: -0.426956282 + -0.625 = -1.051956282 t = -1.051956282 Simplifying t = -1.051956282Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.198043718, -1.051956282}
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