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Simplifying 180 = 16t2 + -130t Reorder the terms: 180 = -130t + 16t2 Solving 180 = -130t + 16t2 Solving for variable 't'. Reorder the terms: 180 + 130t + -16t2 = -130t + 130t + 16t2 + -16t2 Combine like terms: -130t + 130t = 0 180 + 130t + -16t2 = 0 + 16t2 + -16t2 180 + 130t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 180 + 130t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(90 + 65t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(90 + 65t + -8t2)' equal to zero and attempt to solve: Simplifying 90 + 65t + -8t2 = 0 Solving 90 + 65t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -11.25 + -8.125t + t2 = 0 Move the constant term to the right: Add '11.25' to each side of the equation. -11.25 + -8.125t + 11.25 + t2 = 0 + 11.25 Reorder the terms: -11.25 + 11.25 + -8.125t + t2 = 0 + 11.25 Combine like terms: -11.25 + 11.25 = 0.00 0.00 + -8.125t + t2 = 0 + 11.25 -8.125t + t2 = 0 + 11.25 Combine like terms: 0 + 11.25 = 11.25 -8.125t + t2 = 11.25 The t term is -8.125t. Take half its coefficient (-4.0625). Square it (16.50390625) and add it to both sides. Add '16.50390625' to each side of the equation. -8.125t + 16.50390625 + t2 = 11.25 + 16.50390625 Reorder the terms: 16.50390625 + -8.125t + t2 = 11.25 + 16.50390625 Combine like terms: 11.25 + 16.50390625 = 27.75390625 16.50390625 + -8.125t + t2 = 27.75390625 Factor a perfect square on the left side: (t + -4.0625)(t + -4.0625) = 27.75390625 Calculate the square root of the right side: 5.268197628 Break this problem into two subproblems by setting (t + -4.0625) equal to 5.268197628 and -5.268197628.Subproblem 1
t + -4.0625 = 5.268197628 Simplifying t + -4.0625 = 5.268197628 Reorder the terms: -4.0625 + t = 5.268197628 Solving -4.0625 + t = 5.268197628 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.0625' to each side of the equation. -4.0625 + 4.0625 + t = 5.268197628 + 4.0625 Combine like terms: -4.0625 + 4.0625 = 0.0000 0.0000 + t = 5.268197628 + 4.0625 t = 5.268197628 + 4.0625 Combine like terms: 5.268197628 + 4.0625 = 9.330697628 t = 9.330697628 Simplifying t = 9.330697628Subproblem 2
t + -4.0625 = -5.268197628 Simplifying t + -4.0625 = -5.268197628 Reorder the terms: -4.0625 + t = -5.268197628 Solving -4.0625 + t = -5.268197628 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.0625' to each side of the equation. -4.0625 + 4.0625 + t = -5.268197628 + 4.0625 Combine like terms: -4.0625 + 4.0625 = 0.0000 0.0000 + t = -5.268197628 + 4.0625 t = -5.268197628 + 4.0625 Combine like terms: -5.268197628 + 4.0625 = -1.205697628 t = -1.205697628 Simplifying t = -1.205697628Solution
The solution to the problem is based on the solutions from the subproblems. t = {9.330697628, -1.205697628}Solution
t = {9.330697628, -1.205697628}
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