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Simplifying 0 = 12 + 20t + -4t2 Solving 0 = 12 + 20t + -4t2 Solving for variable 't'. Combine like terms: 0 + -12 = -12 -12 + -20t + 4t2 = 12 + 20t + -4t2 + -12 + -20t + 4t2 Reorder the terms: -12 + -20t + 4t2 = 12 + -12 + 20t + -20t + -4t2 + 4t2 Combine like terms: 12 + -12 = 0 -12 + -20t + 4t2 = 0 + 20t + -20t + -4t2 + 4t2 -12 + -20t + 4t2 = 20t + -20t + -4t2 + 4t2 Combine like terms: 20t + -20t = 0 -12 + -20t + 4t2 = 0 + -4t2 + 4t2 -12 + -20t + 4t2 = -4t2 + 4t2 Combine like terms: -4t2 + 4t2 = 0 -12 + -20t + 4t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-3 + -5t + t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-3 + -5t + t2)' equal to zero and attempt to solve: Simplifying -3 + -5t + t2 = 0 Solving -3 + -5t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + -5t + 3 + t2 = 0 + 3 Reorder the terms: -3 + 3 + -5t + t2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + -5t + t2 = 0 + 3 -5t + t2 = 0 + 3 Combine like terms: 0 + 3 = 3 -5t + t2 = 3 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 = 3 + 6.25 Reorder the terms: 6.25 + -5t + t2 = 3 + 6.25 Combine like terms: 3 + 6.25 = 9.25 6.25 + -5t + t2 = 9.25 Factor a perfect square on the left side: (t + -2.5)(t + -2.5) = 9.25 Calculate the square root of the right side: 3.041381265 Break this problem into two subproblems by setting (t + -2.5) equal to 3.041381265 and -3.041381265.Subproblem 1
t + -2.5 = 3.041381265 Simplifying t + -2.5 = 3.041381265 Reorder the terms: -2.5 + t = 3.041381265 Solving -2.5 + t = 3.041381265 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 = 3.041381265 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = 3.041381265 + 2.5 t = 3.041381265 + 2.5 Combine like terms: 3.041381265 + 2.5 = 5.541381265 t = 5.541381265 Simplifying t = 5.541381265Subproblem 2
t + -2.5 = -3.041381265 Simplifying t + -2.5 = -3.041381265 Reorder the terms: -2.5 + t = -3.041381265 Solving -2.5 + t = -3.041381265 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 = -3.041381265 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = -3.041381265 + 2.5 t = -3.041381265 + 2.5 Combine like terms: -3.041381265 + 2.5 = -0.541381265 t = -0.541381265 Simplifying t = -0.541381265Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.541381265, -0.541381265}Solution
t = {5.541381265, -0.541381265}
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