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Simplifying 12t2 + -32t + 15 = 0 Reorder the terms: 15 + -32t + 12t2 = 0 Solving 15 + -32t + 12t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. 1.25 + -2.666666667t + t2 = 0 Move the constant term to the right: Add '-1.25' to each side of the equation. 1.25 + -2.666666667t + -1.25 + t2 = 0 + -1.25 Reorder the terms: 1.25 + -1.25 + -2.666666667t + t2 = 0 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + -2.666666667t + t2 = 0 + -1.25 -2.666666667t + t2 = 0 + -1.25 Combine like terms: 0 + -1.25 = -1.25 -2.666666667t + t2 = -1.25 The t term is -2.666666667t. Take half its coefficient (-1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. -2.666666667t + 1.777777780 + t2 = -1.25 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667t + t2 = -1.25 + 1.777777780 Combine like terms: -1.25 + 1.777777780 = 0.52777778 1.777777780 + -2.666666667t + t2 = 0.52777778 Factor a perfect square on the left side: (t + -1.333333334)(t + -1.333333334) = 0.52777778 Calculate the square root of the right side: 0.726483159 Break this problem into two subproblems by setting (t + -1.333333334) equal to 0.726483159 and -0.726483159.Subproblem 1
t + -1.333333334 = 0.726483159 Simplifying t + -1.333333334 = 0.726483159 Reorder the terms: -1.333333334 + t = 0.726483159 Solving -1.333333334 + t = 0.726483159 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + t = 0.726483159 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + t = 0.726483159 + 1.333333334 t = 0.726483159 + 1.333333334 Combine like terms: 0.726483159 + 1.333333334 = 2.059816493 t = 2.059816493 Simplifying t = 2.059816493Subproblem 2
t + -1.333333334 = -0.726483159 Simplifying t + -1.333333334 = -0.726483159 Reorder the terms: -1.333333334 + t = -0.726483159 Solving -1.333333334 + t = -0.726483159 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + t = -0.726483159 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + t = -0.726483159 + 1.333333334 t = -0.726483159 + 1.333333334 Combine like terms: -0.726483159 + 1.333333334 = 0.606850175 t = 0.606850175 Simplifying t = 0.606850175Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.059816493, 0.606850175}
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