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Simplifying 80t + -16t2 + 10 = 80 Reorder the terms: 10 + 80t + -16t2 = 80 Solving 10 + 80t + -16t2 = 80 Solving for variable 't'. Reorder the terms: 10 + -80 + 80t + -16t2 = 80 + -80 Combine like terms: 10 + -80 = -70 -70 + 80t + -16t2 = 80 + -80 Combine like terms: 80 + -80 = 0 -70 + 80t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-35 + 40t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-35 + 40t + -8t2)' equal to zero and attempt to solve: Simplifying -35 + 40t + -8t2 = 0 Solving -35 + 40t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. 4.375 + -5t + t2 = 0 Move the constant term to the right: Add '-4.375' to each side of the equation. 4.375 + -5t + -4.375 + t2 = 0 + -4.375 Reorder the terms: 4.375 + -4.375 + -5t + t2 = 0 + -4.375 Combine like terms: 4.375 + -4.375 = 0.000 0.000 + -5t + t2 = 0 + -4.375 -5t + t2 = 0 + -4.375 Combine like terms: 0 + -4.375 = -4.375 -5t + t2 = -4.375 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 = -4.375 + 6.25 Reorder the terms: 6.25 + -5t + t2 = -4.375 + 6.25 Combine like terms: -4.375 + 6.25 = 1.875 6.25 + -5t + t2 = 1.875 Factor a perfect square on the left side: (t + -2.5)(t + -2.5) = 1.875 Calculate the square root of the right side: 1.369306394 Break this problem into two subproblems by setting (t + -2.5) equal to 1.369306394 and -1.369306394.Subproblem 1
t + -2.5 = 1.369306394 Simplifying t + -2.5 = 1.369306394 Reorder the terms: -2.5 + t = 1.369306394 Solving -2.5 + t = 1.369306394 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 = 1.369306394 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = 1.369306394 + 2.5 t = 1.369306394 + 2.5 Combine like terms: 1.369306394 + 2.5 = 3.869306394 t = 3.869306394 Simplifying t = 3.869306394Subproblem 2
t + -2.5 = -1.369306394 Simplifying t + -2.5 = -1.369306394 Reorder the terms: -2.5 + t = -1.369306394 Solving -2.5 + t = -1.369306394 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 = -1.369306394 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = -1.369306394 + 2.5 t = -1.369306394 + 2.5 Combine like terms: -1.369306394 + 2.5 = 1.130693606 t = 1.130693606 Simplifying t = 1.130693606Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.869306394, 1.130693606}Solution
t = {3.869306394, 1.130693606}
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