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Simplifying 0 = 6t2 + -10t + -9 Reorder the terms: 0 = -9 + -10t + 6t2 Solving 0 = -9 + -10t + 6t2 Solving for variable 't'. Combine like terms: 0 + 9 = 9 9 + 10t + -6t2 = -9 + -10t + 6t2 + 9 + 10t + -6t2 Reorder the terms: 9 + 10t + -6t2 = -9 + 9 + -10t + 10t + 6t2 + -6t2 Combine like terms: -9 + 9 = 0 9 + 10t + -6t2 = 0 + -10t + 10t + 6t2 + -6t2 9 + 10t + -6t2 = -10t + 10t + 6t2 + -6t2 Combine like terms: -10t + 10t = 0 9 + 10t + -6t2 = 0 + 6t2 + -6t2 9 + 10t + -6t2 = 6t2 + -6t2 Combine like terms: 6t2 + -6t2 = 0 9 + 10t + -6t2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -1.5 + -1.666666667t + t2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -1.666666667t + 1.5 + t2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -1.666666667t + t2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -1.666666667t + t2 = 0 + 1.5 -1.666666667t + t2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -1.666666667t + t2 = 1.5 The t term is -1.666666667t. Take half its coefficient (-0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. -1.666666667t + 0.6944444447 + t2 = 1.5 + 0.6944444447 Reorder the terms: 0.6944444447 + -1.666666667t + t2 = 1.5 + 0.6944444447 Combine like terms: 1.5 + 0.6944444447 = 2.1944444447 0.6944444447 + -1.666666667t + t2 = 2.1944444447 Factor a perfect square on the left side: (t + -0.8333333335)(t + -0.8333333335) = 2.1944444447 Calculate the square root of the right side: 1.481365736 Break this problem into two subproblems by setting (t + -0.8333333335) equal to 1.481365736 and -1.481365736.Subproblem 1
t + -0.8333333335 = 1.481365736 Simplifying t + -0.8333333335 = 1.481365736 Reorder the terms: -0.8333333335 + t = 1.481365736 Solving -0.8333333335 + t = 1.481365736 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + t = 1.481365736 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + t = 1.481365736 + 0.8333333335 t = 1.481365736 + 0.8333333335 Combine like terms: 1.481365736 + 0.8333333335 = 2.3146990695 t = 2.3146990695 Simplifying t = 2.3146990695Subproblem 2
t + -0.8333333335 = -1.481365736 Simplifying t + -0.8333333335 = -1.481365736 Reorder the terms: -0.8333333335 + t = -1.481365736 Solving -0.8333333335 + t = -1.481365736 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + t = -1.481365736 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + t = -1.481365736 + 0.8333333335 t = -1.481365736 + 0.8333333335 Combine like terms: -1.481365736 + 0.8333333335 = -0.6480324025 t = -0.6480324025 Simplifying t = -0.6480324025Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.3146990695, -0.6480324025}
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