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Simplifying 18t2 + -48t + -32 = 0 Reorder the terms: -32 + -48t + 18t2 = 0 Solving -32 + -48t + 18t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(-16 + -24t + 9t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-16 + -24t + 9t2)' equal to zero and attempt to solve: Simplifying -16 + -24t + 9t2 = 0 Solving -16 + -24t + 9t2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1.777777778 + -2.666666667t + t2 = 0 Move the constant term to the right: Add '1.777777778' to each side of the equation. -1.777777778 + -2.666666667t + 1.777777778 + t2 = 0 + 1.777777778 Reorder the terms: -1.777777778 + 1.777777778 + -2.666666667t + t2 = 0 + 1.777777778 Combine like terms: -1.777777778 + 1.777777778 = 0.000000000 0.000000000 + -2.666666667t + t2 = 0 + 1.777777778 -2.666666667t + t2 = 0 + 1.777777778 Combine like terms: 0 + 1.777777778 = 1.777777778 -2.666666667t + t2 = 1.777777778 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 = 1.777777778 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667t + t2 = 1.777777778 + 1.777777780 Combine like terms: 1.777777778 + 1.777777780 = 3.555555558 1.777777780 + -2.666666667t + t2 = 3.555555558 Factor a perfect square on the left side: (t + -1.333333334)(t + -1.333333334) = 3.555555558 Calculate the square root of the right side: 1.885618084 Break this problem into two subproblems by setting (t + -1.333333334) equal to 1.885618084 and -1.885618084.Subproblem 1
t + -1.333333334 = 1.885618084 Simplifying t + -1.333333334 = 1.885618084 Reorder the terms: -1.333333334 + t = 1.885618084 Solving -1.333333334 + t = 1.885618084 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 = 1.885618084 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + t = 1.885618084 + 1.333333334 t = 1.885618084 + 1.333333334 Combine like terms: 1.885618084 + 1.333333334 = 3.218951418 t = 3.218951418 Simplifying t = 3.218951418Subproblem 2
t + -1.333333334 = -1.885618084 Simplifying t + -1.333333334 = -1.885618084 Reorder the terms: -1.333333334 + t = -1.885618084 Solving -1.333333334 + t = -1.885618084 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 = -1.885618084 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + t = -1.885618084 + 1.333333334 t = -1.885618084 + 1.333333334 Combine like terms: -1.885618084 + 1.333333334 = -0.55228475 t = -0.55228475 Simplifying t = -0.55228475Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.218951418, -0.55228475}Solution
t = {3.218951418, -0.55228475}
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