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Simplifying 0 = -5.85t2 + 30t + 6 Reorder the terms: 0 = 6 + 30t + -5.85t2 Solving 0 = 6 + 30t + -5.85t2 Solving for variable 't'. Combine like terms: 0 + -6 = -6 -6 + -30t + 5.85t2 = 6 + 30t + -5.85t2 + -6 + -30t + 5.85t2 Reorder the terms: -6 + -30t + 5.85t2 = 6 + -6 + 30t + -30t + -5.85t2 + 5.85t2 Combine like terms: 6 + -6 = 0 -6 + -30t + 5.85t2 = 0 + 30t + -30t + -5.85t2 + 5.85t2 -6 + -30t + 5.85t2 = 30t + -30t + -5.85t2 + 5.85t2 Combine like terms: 30t + -30t = 0 -6 + -30t + 5.85t2 = 0 + -5.85t2 + 5.85t2 -6 + -30t + 5.85t2 = -5.85t2 + 5.85t2 Combine like terms: -5.85t2 + 5.85t2 = 0.00 -6 + -30t + 5.85t2 = 0.00 Begin completing the square. Divide all terms by 5.85 the coefficient of the squared term: Divide each side by '5.85'. -1.025641026 + -5.128205128t + t2 = 0 Move the constant term to the right: Add '1.025641026' to each side of the equation. -1.025641026 + -5.128205128t + 1.025641026 + t2 = 0 + 1.025641026 Reorder the terms: -1.025641026 + 1.025641026 + -5.128205128t + t2 = 0 + 1.025641026 Combine like terms: -1.025641026 + 1.025641026 = 0.000000000 0.000000000 + -5.128205128t + t2 = 0 + 1.025641026 -5.128205128t + t2 = 0 + 1.025641026 Combine like terms: 0 + 1.025641026 = 1.025641026 -5.128205128t + t2 = 1.025641026 The t term is -5.128205128t. Take half its coefficient (-2.564102564). Square it (6.574621959) and add it to both sides. Add '6.574621959' to each side of the equation. -5.128205128t + 6.574621959 + t2 = 1.025641026 + 6.574621959 Reorder the terms: 6.574621959 + -5.128205128t + t2 = 1.025641026 + 6.574621959 Combine like terms: 1.025641026 + 6.574621959 = 7.600262985 6.574621959 + -5.128205128t + t2 = 7.600262985 Factor a perfect square on the left side: (t + -2.564102564)(t + -2.564102564) = 7.600262985 Calculate the square root of the right side: 2.756857447 Break this problem into two subproblems by setting (t + -2.564102564) equal to 2.756857447 and -2.756857447.Subproblem 1
t + -2.564102564 = 2.756857447 Simplifying t + -2.564102564 = 2.756857447 Reorder the terms: -2.564102564 + t = 2.756857447 Solving -2.564102564 + t = 2.756857447 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.564102564' to each side of the equation. -2.564102564 + 2.564102564 + t = 2.756857447 + 2.564102564 Combine like terms: -2.564102564 + 2.564102564 = 0.000000000 0.000000000 + t = 2.756857447 + 2.564102564 t = 2.756857447 + 2.564102564 Combine like terms: 2.756857447 + 2.564102564 = 5.320960011 t = 5.320960011 Simplifying t = 5.320960011Subproblem 2
t + -2.564102564 = -2.756857447 Simplifying t + -2.564102564 = -2.756857447 Reorder the terms: -2.564102564 + t = -2.756857447 Solving -2.564102564 + t = -2.756857447 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.564102564' to each side of the equation. -2.564102564 + 2.564102564 + t = -2.756857447 + 2.564102564 Combine like terms: -2.564102564 + 2.564102564 = 0.000000000 0.000000000 + t = -2.756857447 + 2.564102564 t = -2.756857447 + 2.564102564 Combine like terms: -2.756857447 + 2.564102564 = -0.192754883 t = -0.192754883 Simplifying t = -0.192754883Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.320960011, -0.192754883}
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