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Simplifying 0 = -16t2 + 22t + 500 Reorder the terms: 0 = 500 + 22t + -16t2 Solving 0 = 500 + 22t + -16t2 Solving for variable 't'. Combine like terms: 0 + -500 = -500 -500 + -22t + 16t2 = 500 + 22t + -16t2 + -500 + -22t + 16t2 Reorder the terms: -500 + -22t + 16t2 = 500 + -500 + 22t + -22t + -16t2 + 16t2 Combine like terms: 500 + -500 = 0 -500 + -22t + 16t2 = 0 + 22t + -22t + -16t2 + 16t2 -500 + -22t + 16t2 = 22t + -22t + -16t2 + 16t2 Combine like terms: 22t + -22t = 0 -500 + -22t + 16t2 = 0 + -16t2 + 16t2 -500 + -22t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -500 + -22t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-250 + -11t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-250 + -11t + 8t2)' equal to zero and attempt to solve: Simplifying -250 + -11t + 8t2 = 0 Solving -250 + -11t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -31.25 + -1.375t + t2 = 0 Move the constant term to the right: Add '31.25' to each side of the equation. -31.25 + -1.375t + 31.25 + t2 = 0 + 31.25 Reorder the terms: -31.25 + 31.25 + -1.375t + t2 = 0 + 31.25 Combine like terms: -31.25 + 31.25 = 0.00 0.00 + -1.375t + t2 = 0 + 31.25 -1.375t + t2 = 0 + 31.25 Combine like terms: 0 + 31.25 = 31.25 -1.375t + t2 = 31.25 The t term is -1.375t. Take half its coefficient (-0.6875). Square it (0.47265625) and add it to both sides. Add '0.47265625' to each side of the equation. -1.375t + 0.47265625 + t2 = 31.25 + 0.47265625 Reorder the terms: 0.47265625 + -1.375t + t2 = 31.25 + 0.47265625 Combine like terms: 31.25 + 0.47265625 = 31.72265625 0.47265625 + -1.375t + t2 = 31.72265625 Factor a perfect square on the left side: (t + -0.6875)(t + -0.6875) = 31.72265625 Calculate the square root of the right side: 5.632286947 Break this problem into two subproblems by setting (t + -0.6875) equal to 5.632286947 and -5.632286947.Subproblem 1
t + -0.6875 = 5.632286947 Simplifying t + -0.6875 = 5.632286947 Reorder the terms: -0.6875 + t = 5.632286947 Solving -0.6875 + t = 5.632286947 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + t = 5.632286947 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + t = 5.632286947 + 0.6875 t = 5.632286947 + 0.6875 Combine like terms: 5.632286947 + 0.6875 = 6.319786947 t = 6.319786947 Simplifying t = 6.319786947Subproblem 2
t + -0.6875 = -5.632286947 Simplifying t + -0.6875 = -5.632286947 Reorder the terms: -0.6875 + t = -5.632286947 Solving -0.6875 + t = -5.632286947 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + t = -5.632286947 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + t = -5.632286947 + 0.6875 t = -5.632286947 + 0.6875 Combine like terms: -5.632286947 + 0.6875 = -4.944786947 t = -4.944786947 Simplifying t = -4.944786947Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.319786947, -4.944786947}Solution
t = {6.319786947, -4.944786947}
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