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Simplifying -16t2 + -18t + 180 = 0 Reorder the terms: 180 + -18t + -16t2 = 0 Solving 180 + -18t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(90 + -9t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(90 + -9t + -8t2)' equal to zero and attempt to solve: Simplifying 90 + -9t + -8t2 = 0 Solving 90 + -9t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -11.25 + 1.125t + t2 = 0 Move the constant term to the right: Add '11.25' to each side of the equation. -11.25 + 1.125t + 11.25 + t2 = 0 + 11.25 Reorder the terms: -11.25 + 11.25 + 1.125t + t2 = 0 + 11.25 Combine like terms: -11.25 + 11.25 = 0.00 0.00 + 1.125t + t2 = 0 + 11.25 1.125t + t2 = 0 + 11.25 Combine like terms: 0 + 11.25 = 11.25 1.125t + t2 = 11.25 The t term is 1.125t. Take half its coefficient (0.5625). Square it (0.31640625) and add it to both sides. Add '0.31640625' to each side of the equation. 1.125t + 0.31640625 + t2 = 11.25 + 0.31640625 Reorder the terms: 0.31640625 + 1.125t + t2 = 11.25 + 0.31640625 Combine like terms: 11.25 + 0.31640625 = 11.56640625 0.31640625 + 1.125t + t2 = 11.56640625 Factor a perfect square on the left side: (t + 0.5625)(t + 0.5625) = 11.56640625 Calculate the square root of the right side: 3.400941965 Break this problem into two subproblems by setting (t + 0.5625) equal to 3.400941965 and -3.400941965.Subproblem 1
t + 0.5625 = 3.400941965 Simplifying t + 0.5625 = 3.400941965 Reorder the terms: 0.5625 + t = 3.400941965 Solving 0.5625 + t = 3.400941965 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5625' to each side of the equation. 0.5625 + -0.5625 + t = 3.400941965 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + t = 3.400941965 + -0.5625 t = 3.400941965 + -0.5625 Combine like terms: 3.400941965 + -0.5625 = 2.838441965 t = 2.838441965 Simplifying t = 2.838441965Subproblem 2
t + 0.5625 = -3.400941965 Simplifying t + 0.5625 = -3.400941965 Reorder the terms: 0.5625 + t = -3.400941965 Solving 0.5625 + t = -3.400941965 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5625' to each side of the equation. 0.5625 + -0.5625 + t = -3.400941965 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + t = -3.400941965 + -0.5625 t = -3.400941965 + -0.5625 Combine like terms: -3.400941965 + -0.5625 = -3.963441965 t = -3.963441965 Simplifying t = -3.963441965Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.838441965, -3.963441965}Solution
t = {2.838441965, -3.963441965}
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