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Simplifying 9t2 + 39t + 36 = 180 Reorder the terms: 36 + 39t + 9t2 = 180 Solving 36 + 39t + 9t2 = 180 Solving for variable 't'. Reorder the terms: 36 + -180 + 39t + 9t2 = 180 + -180 Combine like terms: 36 + -180 = -144 -144 + 39t + 9t2 = 180 + -180 Combine like terms: 180 + -180 = 0 -144 + 39t + 9t2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-48 + 13t + 3t2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-48 + 13t + 3t2)' equal to zero and attempt to solve: Simplifying -48 + 13t + 3t2 = 0 Solving -48 + 13t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -16 + 4.333333333t + t2 = 0 Move the constant term to the right: Add '16' to each side of the equation. -16 + 4.333333333t + 16 + t2 = 0 + 16 Reorder the terms: -16 + 16 + 4.333333333t + t2 = 0 + 16 Combine like terms: -16 + 16 = 0 0 + 4.333333333t + t2 = 0 + 16 4.333333333t + t2 = 0 + 16 Combine like terms: 0 + 16 = 16 4.333333333t + t2 = 16 The t term is 4.333333333t. Take half its coefficient (2.166666667). Square it (4.694444446) and add it to both sides. Add '4.694444446' to each side of the equation. 4.333333333t + 4.694444446 + t2 = 16 + 4.694444446 Reorder the terms: 4.694444446 + 4.333333333t + t2 = 16 + 4.694444446 Combine like terms: 16 + 4.694444446 = 20.694444446 4.694444446 + 4.333333333t + t2 = 20.694444446 Factor a perfect square on the left side: (t + 2.166666667)(t + 2.166666667) = 20.694444446 Calculate the square root of the right side: 4.549114688 Break this problem into two subproblems by setting (t + 2.166666667) equal to 4.549114688 and -4.549114688.Subproblem 1
t + 2.166666667 = 4.549114688 Simplifying t + 2.166666667 = 4.549114688 Reorder the terms: 2.166666667 + t = 4.549114688 Solving 2.166666667 + t = 4.549114688 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.166666667' to each side of the equation. 2.166666667 + -2.166666667 + t = 4.549114688 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + t = 4.549114688 + -2.166666667 t = 4.549114688 + -2.166666667 Combine like terms: 4.549114688 + -2.166666667 = 2.382448021 t = 2.382448021 Simplifying t = 2.382448021Subproblem 2
t + 2.166666667 = -4.549114688 Simplifying t + 2.166666667 = -4.549114688 Reorder the terms: 2.166666667 + t = -4.549114688 Solving 2.166666667 + t = -4.549114688 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.166666667' to each side of the equation. 2.166666667 + -2.166666667 + t = -4.549114688 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + t = -4.549114688 + -2.166666667 t = -4.549114688 + -2.166666667 Combine like terms: -4.549114688 + -2.166666667 = -6.715781355 t = -6.715781355 Simplifying t = -6.715781355Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.382448021, -6.715781355}Solution
t = {2.382448021, -6.715781355}
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