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Simplifying -16t2 + -22t + 100 = 0 Reorder the terms: 100 + -22t + -16t2 = 0 Solving 100 + -22t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(50 + -11t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(50 + -11t + -8t2)' equal to zero and attempt to solve: Simplifying 50 + -11t + -8t2 = 0 Solving 50 + -11t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -6.25 + 1.375t + t2 = 0 Move the constant term to the right: Add '6.25' to each side of the equation. -6.25 + 1.375t + 6.25 + t2 = 0 + 6.25 Reorder the terms: -6.25 + 6.25 + 1.375t + t2 = 0 + 6.25 Combine like terms: -6.25 + 6.25 = 0.00 0.00 + 1.375t + t2 = 0 + 6.25 1.375t + t2 = 0 + 6.25 Combine like terms: 0 + 6.25 = 6.25 1.375t + t2 = 6.25 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 = 6.25 + 0.47265625 Reorder the terms: 0.47265625 + 1.375t + t2 = 6.25 + 0.47265625 Combine like terms: 6.25 + 0.47265625 = 6.72265625 0.47265625 + 1.375t + t2 = 6.72265625 Factor a perfect square on the left side: (t + 0.6875)(t + 0.6875) = 6.72265625 Calculate the square root of the right side: 2.592808564 Break this problem into two subproblems by setting (t + 0.6875) equal to 2.592808564 and -2.592808564.Subproblem 1
t + 0.6875 = 2.592808564 Simplifying t + 0.6875 = 2.592808564 Reorder the terms: 0.6875 + t = 2.592808564 Solving 0.6875 + t = 2.592808564 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 = 2.592808564 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + t = 2.592808564 + -0.6875 t = 2.592808564 + -0.6875 Combine like terms: 2.592808564 + -0.6875 = 1.905308564 t = 1.905308564 Simplifying t = 1.905308564Subproblem 2
t + 0.6875 = -2.592808564 Simplifying t + 0.6875 = -2.592808564 Reorder the terms: 0.6875 + t = -2.592808564 Solving 0.6875 + t = -2.592808564 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 = -2.592808564 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + t = -2.592808564 + -0.6875 t = -2.592808564 + -0.6875 Combine like terms: -2.592808564 + -0.6875 = -3.280308564 t = -3.280308564 Simplifying t = -3.280308564Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.905308564, -3.280308564}Solution
t = {1.905308564, -3.280308564}
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