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Simplifying 3t2 + -12t + 9 = 1 Reorder the terms: 9 + -12t + 3t2 = 1 Solving 9 + -12t + 3t2 = 1 Solving for variable 't'. Reorder the terms: 9 + -1 + -12t + 3t2 = 1 + -1 Combine like terms: 9 + -1 = 8 8 + -12t + 3t2 = 1 + -1 Combine like terms: 1 + -1 = 0 8 + -12t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 2.666666667 + -4t + t2 = 0 Move the constant term to the right: Add '-2.666666667' to each side of the equation. 2.666666667 + -4t + -2.666666667 + t2 = 0 + -2.666666667 Reorder the terms: 2.666666667 + -2.666666667 + -4t + t2 = 0 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + -4t + t2 = 0 + -2.666666667 -4t + t2 = 0 + -2.666666667 Combine like terms: 0 + -2.666666667 = -2.666666667 -4t + t2 = -2.666666667 The t term is -4t. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4t + 4 + t2 = -2.666666667 + 4 Reorder the terms: 4 + -4t + t2 = -2.666666667 + 4 Combine like terms: -2.666666667 + 4 = 1.333333333 4 + -4t + t2 = 1.333333333 Factor a perfect square on the left side: (t + -2)(t + -2) = 1.333333333 Calculate the square root of the right side: 1.154700538 Break this problem into two subproblems by setting (t + -2) equal to 1.154700538 and -1.154700538.Subproblem 1
t + -2 = 1.154700538 Simplifying t + -2 = 1.154700538 Reorder the terms: -2 + t = 1.154700538 Solving -2 + t = 1.154700538 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = 1.154700538 + 2 Combine like terms: -2 + 2 = 0 0 + t = 1.154700538 + 2 t = 1.154700538 + 2 Combine like terms: 1.154700538 + 2 = 3.154700538 t = 3.154700538 Simplifying t = 3.154700538Subproblem 2
t + -2 = -1.154700538 Simplifying t + -2 = -1.154700538 Reorder the terms: -2 + t = -1.154700538 Solving -2 + t = -1.154700538 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = -1.154700538 + 2 Combine like terms: -2 + 2 = 0 0 + t = -1.154700538 + 2 t = -1.154700538 + 2 Combine like terms: -1.154700538 + 2 = 0.845299462 t = 0.845299462 Simplifying t = 0.845299462Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.154700538, 0.845299462}
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