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Simplifying 20 = 45t + -5t2 Solving 20 = 45t + -5t2 Solving for variable 't'. Reorder the terms: 20 + -45t + 5t2 = 45t + -45t + -5t2 + 5t2 Combine like terms: 45t + -45t = 0 20 + -45t + 5t2 = 0 + -5t2 + 5t2 20 + -45t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 20 + -45t + 5t2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(4 + -9t + t2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(4 + -9t + t2)' equal to zero and attempt to solve: Simplifying 4 + -9t + t2 = 0 Solving 4 + -9t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-4' to each side of the equation. 4 + -9t + -4 + t2 = 0 + -4 Reorder the terms: 4 + -4 + -9t + t2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -9t + t2 = 0 + -4 -9t + t2 = 0 + -4 Combine like terms: 0 + -4 = -4 -9t + t2 = -4 The t term is -9t. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9t + 20.25 + t2 = -4 + 20.25 Reorder the terms: 20.25 + -9t + t2 = -4 + 20.25 Combine like terms: -4 + 20.25 = 16.25 20.25 + -9t + t2 = 16.25 Factor a perfect square on the left side: (t + -4.5)(t + -4.5) = 16.25 Calculate the square root of the right side: 4.031128874 Break this problem into two subproblems by setting (t + -4.5) equal to 4.031128874 and -4.031128874.Subproblem 1
t + -4.5 = 4.031128874 Simplifying t + -4.5 = 4.031128874 Reorder the terms: -4.5 + t = 4.031128874 Solving -4.5 + t = 4.031128874 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + t = 4.031128874 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = 4.031128874 + 4.5 t = 4.031128874 + 4.5 Combine like terms: 4.031128874 + 4.5 = 8.531128874 t = 8.531128874 Simplifying t = 8.531128874Subproblem 2
t + -4.5 = -4.031128874 Simplifying t + -4.5 = -4.031128874 Reorder the terms: -4.5 + t = -4.031128874 Solving -4.5 + t = -4.031128874 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + t = -4.031128874 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = -4.031128874 + 4.5 t = -4.031128874 + 4.5 Combine like terms: -4.031128874 + 4.5 = 0.468871126 t = 0.468871126 Simplifying t = 0.468871126Solution
The solution to the problem is based on the solutions from the subproblems. t = {8.531128874, 0.468871126}Solution
t = {8.531128874, 0.468871126}
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