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Simplifying 20t + 4t2 = 100 Solving 20t + 4t2 = 100 Solving for variable 't'. Reorder the terms: -100 + 20t + 4t2 = 100 + -100 Combine like terms: 100 + -100 = 0 -100 + 20t + 4t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-25 + 5t + t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-25 + 5t + t2)' equal to zero and attempt to solve: Simplifying -25 + 5t + t2 = 0 Solving -25 + 5t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '25' to each side of the equation. -25 + 5t + 25 + t2 = 0 + 25 Reorder the terms: -25 + 25 + 5t + t2 = 0 + 25 Combine like terms: -25 + 25 = 0 0 + 5t + t2 = 0 + 25 5t + t2 = 0 + 25 Combine like terms: 0 + 25 = 25 5t + t2 = 25 The t term is 5t. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5t + 6.25 + t2 = 25 + 6.25 Reorder the terms: 6.25 + 5t + t2 = 25 + 6.25 Combine like terms: 25 + 6.25 = 31.25 6.25 + 5t + t2 = 31.25 Factor a perfect square on the left side: (t + 2.5)(t + 2.5) = 31.25 Calculate the square root of the right side: 5.590169944 Break this problem into two subproblems by setting (t + 2.5) equal to 5.590169944 and -5.590169944.Subproblem 1
t + 2.5 = 5.590169944 Simplifying t + 2.5 = 5.590169944 Reorder the terms: 2.5 + t = 5.590169944 Solving 2.5 + t = 5.590169944 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + t = 5.590169944 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = 5.590169944 + -2.5 t = 5.590169944 + -2.5 Combine like terms: 5.590169944 + -2.5 = 3.090169944 t = 3.090169944 Simplifying t = 3.090169944Subproblem 2
t + 2.5 = -5.590169944 Simplifying t + 2.5 = -5.590169944 Reorder the terms: 2.5 + t = -5.590169944 Solving 2.5 + t = -5.590169944 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + t = -5.590169944 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = -5.590169944 + -2.5 t = -5.590169944 + -2.5 Combine like terms: -5.590169944 + -2.5 = -8.090169944 t = -8.090169944 Simplifying t = -8.090169944Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.090169944, -8.090169944}Solution
t = {3.090169944, -8.090169944}
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