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Simplifying 30 = -16t2 + 42t + 6 Reorder the terms: 30 = 6 + 42t + -16t2 Solving 30 = 6 + 42t + -16t2 Solving for variable 't'. Combine like terms: 30 + -6 = 24 24 + -42t + 16t2 = 6 + 42t + -16t2 + -6 + -42t + 16t2 Reorder the terms: 24 + -42t + 16t2 = 6 + -6 + 42t + -42t + -16t2 + 16t2 Combine like terms: 6 + -6 = 0 24 + -42t + 16t2 = 0 + 42t + -42t + -16t2 + 16t2 24 + -42t + 16t2 = 42t + -42t + -16t2 + 16t2 Combine like terms: 42t + -42t = 0 24 + -42t + 16t2 = 0 + -16t2 + 16t2 24 + -42t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 24 + -42t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(12 + -21t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(12 + -21t + 8t2)' equal to zero and attempt to solve: Simplifying 12 + -21t + 8t2 = 0 Solving 12 + -21t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 1.5 + -2.625t + t2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -2.625t + -1.5 + t2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -2.625t + t2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -2.625t + t2 = 0 + -1.5 -2.625t + t2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -2.625t + t2 = -1.5 The t term is -2.625t. Take half its coefficient (-1.3125). Square it (1.72265625) and add it to both sides. Add '1.72265625' to each side of the equation. -2.625t + 1.72265625 + t2 = -1.5 + 1.72265625 Reorder the terms: 1.72265625 + -2.625t + t2 = -1.5 + 1.72265625 Combine like terms: -1.5 + 1.72265625 = 0.22265625 1.72265625 + -2.625t + t2 = 0.22265625 Factor a perfect square on the left side: (t + -1.3125)(t + -1.3125) = 0.22265625 Calculate the square root of the right side: 0.471864652 Break this problem into two subproblems by setting (t + -1.3125) equal to 0.471864652 and -0.471864652.Subproblem 1
t + -1.3125 = 0.471864652 Simplifying t + -1.3125 = 0.471864652 Reorder the terms: -1.3125 + t = 0.471864652 Solving -1.3125 + t = 0.471864652 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.3125' to each side of the equation. -1.3125 + 1.3125 + t = 0.471864652 + 1.3125 Combine like terms: -1.3125 + 1.3125 = 0.0000 0.0000 + t = 0.471864652 + 1.3125 t = 0.471864652 + 1.3125 Combine like terms: 0.471864652 + 1.3125 = 1.784364652 t = 1.784364652 Simplifying t = 1.784364652Subproblem 2
t + -1.3125 = -0.471864652 Simplifying t + -1.3125 = -0.471864652 Reorder the terms: -1.3125 + t = -0.471864652 Solving -1.3125 + t = -0.471864652 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.3125' to each side of the equation. -1.3125 + 1.3125 + t = -0.471864652 + 1.3125 Combine like terms: -1.3125 + 1.3125 = 0.0000 0.0000 + t = -0.471864652 + 1.3125 t = -0.471864652 + 1.3125 Combine like terms: -0.471864652 + 1.3125 = 0.840635348 t = 0.840635348 Simplifying t = 0.840635348Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.784364652, 0.840635348}Solution
t = {1.784364652, 0.840635348}
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