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Simplifying 6 = 1 + 25t + -5t2 Solving 6 = 1 + 25t + -5t2 Solving for variable 't'. Combine like terms: 6 + -1 = 5 5 + -25t + 5t2 = 1 + 25t + -5t2 + -1 + -25t + 5t2 Reorder the terms: 5 + -25t + 5t2 = 1 + -1 + 25t + -25t + -5t2 + 5t2 Combine like terms: 1 + -1 = 0 5 + -25t + 5t2 = 0 + 25t + -25t + -5t2 + 5t2 5 + -25t + 5t2 = 25t + -25t + -5t2 + 5t2 Combine like terms: 25t + -25t = 0 5 + -25t + 5t2 = 0 + -5t2 + 5t2 5 + -25t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 5 + -25t + 5t2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(1 + -5t + t2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(1 + -5t + t2)' equal to zero and attempt to solve: Simplifying 1 + -5t + t2 = 0 Solving 1 + -5t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + -5t + -1 + t2 = 0 + -1 Reorder the terms: 1 + -1 + -5t + t2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -5t + t2 = 0 + -1 -5t + t2 = 0 + -1 Combine like terms: 0 + -1 = -1 -5t + t2 = -1 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 = -1 + 6.25 Reorder the terms: 6.25 + -5t + t2 = -1 + 6.25 Combine like terms: -1 + 6.25 = 5.25 6.25 + -5t + t2 = 5.25 Factor a perfect square on the left side: (t + -2.5)(t + -2.5) = 5.25 Calculate the square root of the right side: 2.291287847 Break this problem into two subproblems by setting (t + -2.5) equal to 2.291287847 and -2.291287847.Subproblem 1
t + -2.5 = 2.291287847 Simplifying t + -2.5 = 2.291287847 Reorder the terms: -2.5 + t = 2.291287847 Solving -2.5 + t = 2.291287847 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 = 2.291287847 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = 2.291287847 + 2.5 t = 2.291287847 + 2.5 Combine like terms: 2.291287847 + 2.5 = 4.791287847 t = 4.791287847 Simplifying t = 4.791287847Subproblem 2
t + -2.5 = -2.291287847 Simplifying t + -2.5 = -2.291287847 Reorder the terms: -2.5 + t = -2.291287847 Solving -2.5 + t = -2.291287847 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 = -2.291287847 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = -2.291287847 + 2.5 t = -2.291287847 + 2.5 Combine like terms: -2.291287847 + 2.5 = 0.208712153 t = 0.208712153 Simplifying t = 0.208712153Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.791287847, 0.208712153}Solution
t = {4.791287847, 0.208712153}
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