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Simplifying 0.6t2 + t + -0.5 = 0 Reorder the terms: -0.5 + t + 0.6t2 = 0 Solving -0.5 + t + 0.6t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 0.6 the coefficient of the squared term: Divide each side by '0.6'. -0.8333333333 + 1.666666667t + t2 = 0 Move the constant term to the right: Add '0.8333333333' to each side of the equation. -0.8333333333 + 1.666666667t + 0.8333333333 + t2 = 0 + 0.8333333333 Reorder the terms: -0.8333333333 + 0.8333333333 + 1.666666667t + t2 = 0 + 0.8333333333 Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000 0.0000000000 + 1.666666667t + t2 = 0 + 0.8333333333 1.666666667t + t2 = 0 + 0.8333333333 Combine like terms: 0 + 0.8333333333 = 0.8333333333 1.666666667t + t2 = 0.8333333333 The t term is t. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 1.666666667t + 0.25 + t2 = 0.8333333333 + 0.25 Reorder the terms: 0.25 + 1.666666667t + t2 = 0.8333333333 + 0.25 Combine like terms: 0.8333333333 + 0.25 = 1.0833333333 0.25 + 1.666666667t + t2 = 1.0833333333 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 1.0833333333 Calculate the square root of the right side: 1.040833 Break this problem into two subproblems by setting (t + 0.5) equal to 1.040833 and -1.040833.Subproblem 1
t + 0.5 = 1.040833 Simplifying t + 0.5 = 1.040833 Reorder the terms: 0.5 + t = 1.040833 Solving 0.5 + t = 1.040833 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = 1.040833 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 1.040833 + -0.5 t = 1.040833 + -0.5 Combine like terms: 1.040833 + -0.5 = 0.540833 t = 0.540833 Simplifying t = 0.540833Subproblem 2
t + 0.5 = -1.040833 Simplifying t + 0.5 = -1.040833 Reorder the terms: 0.5 + t = -1.040833 Solving 0.5 + t = -1.040833 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = -1.040833 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -1.040833 + -0.5 t = -1.040833 + -0.5 Combine like terms: -1.040833 + -0.5 = -1.540833 t = -1.540833 Simplifying t = -1.540833Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.540833, -1.540833}
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