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Simplifying -2t2 + 12t + 9 = 6 Reorder the terms: 9 + 12t + -2t2 = 6 Solving 9 + 12t + -2t2 = 6 Solving for variable 't'. Reorder the terms: 9 + -6 + 12t + -2t2 = 6 + -6 Combine like terms: 9 + -6 = 3 3 + 12t + -2t2 = 6 + -6 Combine like terms: 6 + -6 = 0 3 + 12t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -1.5 + -6t + t2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -6t + 1.5 + t2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -6t + t2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -6t + t2 = 0 + 1.5 -6t + t2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -6t + t2 = 1.5 The t term is -6t. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6t + 9 + t2 = 1.5 + 9 Reorder the terms: 9 + -6t + t2 = 1.5 + 9 Combine like terms: 1.5 + 9 = 10.5 9 + -6t + t2 = 10.5 Factor a perfect square on the left side: (t + -3)(t + -3) = 10.5 Calculate the square root of the right side: 3.240370349 Break this problem into two subproblems by setting (t + -3) equal to 3.240370349 and -3.240370349.Subproblem 1
t + -3 = 3.240370349 Simplifying t + -3 = 3.240370349 Reorder the terms: -3 + t = 3.240370349 Solving -3 + t = 3.240370349 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = 3.240370349 + 3 Combine like terms: -3 + 3 = 0 0 + t = 3.240370349 + 3 t = 3.240370349 + 3 Combine like terms: 3.240370349 + 3 = 6.240370349 t = 6.240370349 Simplifying t = 6.240370349Subproblem 2
t + -3 = -3.240370349 Simplifying t + -3 = -3.240370349 Reorder the terms: -3 + t = -3.240370349 Solving -3 + t = -3.240370349 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = -3.240370349 + 3 Combine like terms: -3 + 3 = 0 0 + t = -3.240370349 + 3 t = -3.240370349 + 3 Combine like terms: -3.240370349 + 3 = -0.240370349 t = -0.240370349 Simplifying t = -0.240370349Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.240370349, -0.240370349}
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