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Simplifying -16t2 + 68t + 5 = 11 Reorder the terms: 5 + 68t + -16t2 = 11 Solving 5 + 68t + -16t2 = 11 Solving for variable 't'. Reorder the terms: 5 + -11 + 68t + -16t2 = 11 + -11 Combine like terms: 5 + -11 = -6 -6 + 68t + -16t2 = 11 + -11 Combine like terms: 11 + -11 = 0 -6 + 68t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + 34t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + 34t + -8t2)' equal to zero and attempt to solve: Simplifying -3 + 34t + -8t2 = 0 Solving -3 + 34t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. 0.375 + -4.25t + t2 = 0 Move the constant term to the right: Add '-0.375' to each side of the equation. 0.375 + -4.25t + -0.375 + t2 = 0 + -0.375 Reorder the terms: 0.375 + -0.375 + -4.25t + t2 = 0 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + -4.25t + t2 = 0 + -0.375 -4.25t + t2 = 0 + -0.375 Combine like terms: 0 + -0.375 = -0.375 -4.25t + t2 = -0.375 The t term is -4.25t. Take half its coefficient (-2.125). Square it (4.515625) and add it to both sides. Add '4.515625' to each side of the equation. -4.25t + 4.515625 + t2 = -0.375 + 4.515625 Reorder the terms: 4.515625 + -4.25t + t2 = -0.375 + 4.515625 Combine like terms: -0.375 + 4.515625 = 4.140625 4.515625 + -4.25t + t2 = 4.140625 Factor a perfect square on the left side: (t + -2.125)(t + -2.125) = 4.140625 Calculate the square root of the right side: 2.034852575 Break this problem into two subproblems by setting (t + -2.125) equal to 2.034852575 and -2.034852575.Subproblem 1
t + -2.125 = 2.034852575 Simplifying t + -2.125 = 2.034852575 Reorder the terms: -2.125 + t = 2.034852575 Solving -2.125 + t = 2.034852575 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.125' to each side of the equation. -2.125 + 2.125 + t = 2.034852575 + 2.125 Combine like terms: -2.125 + 2.125 = 0.000 0.000 + t = 2.034852575 + 2.125 t = 2.034852575 + 2.125 Combine like terms: 2.034852575 + 2.125 = 4.159852575 t = 4.159852575 Simplifying t = 4.159852575Subproblem 2
t + -2.125 = -2.034852575 Simplifying t + -2.125 = -2.034852575 Reorder the terms: -2.125 + t = -2.034852575 Solving -2.125 + t = -2.034852575 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.125' to each side of the equation. -2.125 + 2.125 + t = -2.034852575 + 2.125 Combine like terms: -2.125 + 2.125 = 0.000 0.000 + t = -2.034852575 + 2.125 t = -2.034852575 + 2.125 Combine like terms: -2.034852575 + 2.125 = 0.090147425 t = 0.090147425 Simplifying t = 0.090147425Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.159852575, 0.090147425}Solution
t = {4.159852575, 0.090147425}
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