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Simplifying t2 + -59t + 54 + -82t2 + 60t = 0 Reorder the terms: 54 + -59t + 60t + t2 + -82t2 = 0 Combine like terms: -59t + 60t = 1t 54 + 1t + t2 + -82t2 = 0 Combine like terms: t2 + -82t2 = -81t2 54 + 1t + -81t2 = 0 Solving 54 + 1t + -81t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -81 the coefficient of the squared term: Divide each side by '-81'. -0.6666666667 + -0.01234567901t + t2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + -0.01234567901t + 0.6666666667 + t2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + -0.01234567901t + t2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + -0.01234567901t + t2 = 0 + 0.6666666667 -0.01234567901t + t2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 -0.01234567901t + t2 = 0.6666666667 The t term is -0.01234567901t. Take half its coefficient (-0.006172839505). Square it (0.00003810394755) and add it to both sides. Add '0.00003810394755' to each side of the equation. -0.01234567901t + 0.00003810394755 + t2 = 0.6666666667 + 0.00003810394755 Reorder the terms: 0.00003810394755 + -0.01234567901t + t2 = 0.6666666667 + 0.00003810394755 Combine like terms: 0.6666666667 + 0.00003810394755 = 0.66670477064755 0.00003810394755 + -0.01234567901t + t2 = 0.66670477064755 Factor a perfect square on the left side: (t + -0.006172839505)(t + -0.006172839505) = 0.66670477064755 Calculate the square root of the right side: 0.816519914 Break this problem into two subproblems by setting (t + -0.006172839505) equal to 0.816519914 and -0.816519914.Subproblem 1
t + -0.006172839505 = 0.816519914 Simplifying t + -0.006172839505 = 0.816519914 Reorder the terms: -0.006172839505 + t = 0.816519914 Solving -0.006172839505 + t = 0.816519914 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.006172839505' to each side of the equation. -0.006172839505 + 0.006172839505 + t = 0.816519914 + 0.006172839505 Combine like terms: -0.006172839505 + 0.006172839505 = 0.000000000000 0.000000000000 + t = 0.816519914 + 0.006172839505 t = 0.816519914 + 0.006172839505 Combine like terms: 0.816519914 + 0.006172839505 = 0.822692753505 t = 0.822692753505 Simplifying t = 0.822692753505Subproblem 2
t + -0.006172839505 = -0.816519914 Simplifying t + -0.006172839505 = -0.816519914 Reorder the terms: -0.006172839505 + t = -0.816519914 Solving -0.006172839505 + t = -0.816519914 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.006172839505' to each side of the equation. -0.006172839505 + 0.006172839505 + t = -0.816519914 + 0.006172839505 Combine like terms: -0.006172839505 + 0.006172839505 = 0.000000000000 0.000000000000 + t = -0.816519914 + 0.006172839505 t = -0.816519914 + 0.006172839505 Combine like terms: -0.816519914 + 0.006172839505 = -0.810347074495 t = -0.810347074495 Simplifying t = -0.810347074495Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.822692753505, -0.810347074495}
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