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Simplifying 12 = -16t2 + 22t + 20 Reorder the terms: 12 = 20 + 22t + -16t2 Solving 12 = 20 + 22t + -16t2 Solving for variable 't'. Combine like terms: 12 + -20 = -8 -8 + -22t + 16t2 = 20 + 22t + -16t2 + -20 + -22t + 16t2 Reorder the terms: -8 + -22t + 16t2 = 20 + -20 + 22t + -22t + -16t2 + 16t2 Combine like terms: 20 + -20 = 0 -8 + -22t + 16t2 = 0 + 22t + -22t + -16t2 + 16t2 -8 + -22t + 16t2 = 22t + -22t + -16t2 + 16t2 Combine like terms: 22t + -22t = 0 -8 + -22t + 16t2 = 0 + -16t2 + 16t2 -8 + -22t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -8 + -22t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-4 + -11t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-4 + -11t + 8t2)' equal to zero and attempt to solve: Simplifying -4 + -11t + 8t2 = 0 Solving -4 + -11t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.5 + -1.375t + t2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -1.375t + 0.5 + t2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -1.375t + t2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -1.375t + t2 = 0 + 0.5 -1.375t + t2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -1.375t + t2 = 0.5 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 = 0.5 + 0.47265625 Reorder the terms: 0.47265625 + -1.375t + t2 = 0.5 + 0.47265625 Combine like terms: 0.5 + 0.47265625 = 0.97265625 0.47265625 + -1.375t + t2 = 0.97265625 Factor a perfect square on the left side: (t + -0.6875)(t + -0.6875) = 0.97265625 Calculate the square root of the right side: 0.986233365 Break this problem into two subproblems by setting (t + -0.6875) equal to 0.986233365 and -0.986233365.Subproblem 1
t + -0.6875 = 0.986233365 Simplifying t + -0.6875 = 0.986233365 Reorder the terms: -0.6875 + t = 0.986233365 Solving -0.6875 + t = 0.986233365 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 = 0.986233365 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + t = 0.986233365 + 0.6875 t = 0.986233365 + 0.6875 Combine like terms: 0.986233365 + 0.6875 = 1.673733365 t = 1.673733365 Simplifying t = 1.673733365Subproblem 2
t + -0.6875 = -0.986233365 Simplifying t + -0.6875 = -0.986233365 Reorder the terms: -0.6875 + t = -0.986233365 Solving -0.6875 + t = -0.986233365 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 = -0.986233365 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + t = -0.986233365 + 0.6875 t = -0.986233365 + 0.6875 Combine like terms: -0.986233365 + 0.6875 = -0.298733365 t = -0.298733365 Simplifying t = -0.298733365Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.673733365, -0.298733365}Solution
t = {1.673733365, -0.298733365}
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