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Simplifying 6t2 + -18t + 5 = 0 Reorder the terms: 5 + -18t + 6t2 = 0 Solving 5 + -18t + 6t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.8333333333 + -3t + t2 = 0 Move the constant term to the right: Add '-0.8333333333' to each side of the equation. 0.8333333333 + -3t + -0.8333333333 + t2 = 0 + -0.8333333333 Reorder the terms: 0.8333333333 + -0.8333333333 + -3t + t2 = 0 + -0.8333333333 Combine like terms: 0.8333333333 + -0.8333333333 = 0.0000000000 0.0000000000 + -3t + t2 = 0 + -0.8333333333 -3t + t2 = 0 + -0.8333333333 Combine like terms: 0 + -0.8333333333 = -0.8333333333 -3t + t2 = -0.8333333333 The t term is -3t. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3t + 2.25 + t2 = -0.8333333333 + 2.25 Reorder the terms: 2.25 + -3t + t2 = -0.8333333333 + 2.25 Combine like terms: -0.8333333333 + 2.25 = 1.4166666667 2.25 + -3t + t2 = 1.4166666667 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 1.4166666667 Calculate the square root of the right side: 1.190238071 Break this problem into two subproblems by setting (t + -1.5) equal to 1.190238071 and -1.190238071.Subproblem 1
t + -1.5 = 1.190238071 Simplifying t + -1.5 = 1.190238071 Reorder the terms: -1.5 + t = 1.190238071 Solving -1.5 + t = 1.190238071 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = 1.190238071 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 1.190238071 + 1.5 t = 1.190238071 + 1.5 Combine like terms: 1.190238071 + 1.5 = 2.690238071 t = 2.690238071 Simplifying t = 2.690238071Subproblem 2
t + -1.5 = -1.190238071 Simplifying t + -1.5 = -1.190238071 Reorder the terms: -1.5 + t = -1.190238071 Solving -1.5 + t = -1.190238071 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = -1.190238071 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -1.190238071 + 1.5 t = -1.190238071 + 1.5 Combine like terms: -1.190238071 + 1.5 = 0.309761929 t = 0.309761929 Simplifying t = 0.309761929Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.690238071, 0.309761929}
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