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Simplifying 16t2 + 4t + -80 = 0 Reorder the terms: -80 + 4t + 16t2 = 0 Solving -80 + 4t + 16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '4'. 4(-20 + t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-20 + t + 4t2)' equal to zero and attempt to solve: Simplifying -20 + t + 4t2 = 0 Solving -20 + t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -5 + 0.25t + t2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + 0.25t + 5 + t2 = 0 + 5 Reorder the terms: -5 + 5 + 0.25t + t2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 0.25t + t2 = 0 + 5 0.25t + t2 = 0 + 5 Combine like terms: 0 + 5 = 5 0.25t + t2 = 5 The t term is t. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.25t + 0.25 + t2 = 5 + 0.25 Reorder the terms: 0.25 + 0.25t + t2 = 5 + 0.25 Combine like terms: 5 + 0.25 = 5.25 0.25 + 0.25t + t2 = 5.25 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 5.25 Calculate the square root of the right side: 2.291287847 Break this problem into two subproblems by setting (t + 0.5) equal to 2.291287847 and -2.291287847.Subproblem 1
t + 0.5 = 2.291287847 Simplifying t + 0.5 = 2.291287847 Reorder the terms: 0.5 + t = 2.291287847 Solving 0.5 + t = 2.291287847 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = 2.291287847 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 2.291287847 + -0.5 t = 2.291287847 + -0.5 Combine like terms: 2.291287847 + -0.5 = 1.791287847 t = 1.791287847 Simplifying t = 1.791287847Subproblem 2
t + 0.5 = -2.291287847 Simplifying t + 0.5 = -2.291287847 Reorder the terms: 0.5 + t = -2.291287847 Solving 0.5 + t = -2.291287847 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = -2.291287847 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -2.291287847 + -0.5 t = -2.291287847 + -0.5 Combine like terms: -2.291287847 + -0.5 = -2.791287847 t = -2.791287847 Simplifying t = -2.791287847Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.791287847, -2.791287847}Solution
t = {1.791287847, -2.791287847}
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