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Simplifying -3t2 + 14t + 12 = 0 Reorder the terms: 12 + 14t + -3t2 = 0 Solving 12 + 14t + -3t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -4 + -4.666666667t + t2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + -4.666666667t + 4 + t2 = 0 + 4 Reorder the terms: -4 + 4 + -4.666666667t + t2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -4.666666667t + t2 = 0 + 4 -4.666666667t + t2 = 0 + 4 Combine like terms: 0 + 4 = 4 -4.666666667t + t2 = 4 The t term is -4.666666667t. Take half its coefficient (-2.333333334). Square it (5.444444448) and add it to both sides. Add '5.444444448' to each side of the equation. -4.666666667t + 5.444444448 + t2 = 4 + 5.444444448 Reorder the terms: 5.444444448 + -4.666666667t + t2 = 4 + 5.444444448 Combine like terms: 4 + 5.444444448 = 9.444444448 5.444444448 + -4.666666667t + t2 = 9.444444448 Factor a perfect square on the left side: (t + -2.333333334)(t + -2.333333334) = 9.444444448 Calculate the square root of the right side: 3.073181486 Break this problem into two subproblems by setting (t + -2.333333334) equal to 3.073181486 and -3.073181486.Subproblem 1
t + -2.333333334 = 3.073181486 Simplifying t + -2.333333334 = 3.073181486 Reorder the terms: -2.333333334 + t = 3.073181486 Solving -2.333333334 + t = 3.073181486 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.333333334' to each side of the equation. -2.333333334 + 2.333333334 + t = 3.073181486 + 2.333333334 Combine like terms: -2.333333334 + 2.333333334 = 0.000000000 0.000000000 + t = 3.073181486 + 2.333333334 t = 3.073181486 + 2.333333334 Combine like terms: 3.073181486 + 2.333333334 = 5.40651482 t = 5.40651482 Simplifying t = 5.40651482Subproblem 2
t + -2.333333334 = -3.073181486 Simplifying t + -2.333333334 = -3.073181486 Reorder the terms: -2.333333334 + t = -3.073181486 Solving -2.333333334 + t = -3.073181486 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.333333334' to each side of the equation. -2.333333334 + 2.333333334 + t = -3.073181486 + 2.333333334 Combine like terms: -2.333333334 + 2.333333334 = 0.000000000 0.000000000 + t = -3.073181486 + 2.333333334 t = -3.073181486 + 2.333333334 Combine like terms: -3.073181486 + 2.333333334 = -0.739848152 t = -0.739848152 Simplifying t = -0.739848152Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.40651482, -0.739848152}
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