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Simplifying -16t2 + 24t + 1 = 0 Reorder the terms: 1 + 24t + -16t2 = 0 Solving 1 + 24t + -16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.0625 + -1.5t + t2 = 0 Move the constant term to the right: Add '0.0625' to each side of the equation. -0.0625 + -1.5t + 0.0625 + t2 = 0 + 0.0625 Reorder the terms: -0.0625 + 0.0625 + -1.5t + t2 = 0 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + -1.5t + t2 = 0 + 0.0625 -1.5t + t2 = 0 + 0.0625 Combine like terms: 0 + 0.0625 = 0.0625 -1.5t + t2 = 0.0625 The t term is -1.5t. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5t + 0.5625 + t2 = 0.0625 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 0.0625 + 0.5625 Combine like terms: 0.0625 + 0.5625 = 0.625 0.5625 + -1.5t + t2 = 0.625 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 0.625 Calculate the square root of the right side: 0.790569415 Break this problem into two subproblems by setting (t + -0.75) equal to 0.790569415 and -0.790569415.Subproblem 1
t + -0.75 = 0.790569415 Simplifying t + -0.75 = 0.790569415 Reorder the terms: -0.75 + t = 0.790569415 Solving -0.75 + t = 0.790569415 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = 0.790569415 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 0.790569415 + 0.75 t = 0.790569415 + 0.75 Combine like terms: 0.790569415 + 0.75 = 1.540569415 t = 1.540569415 Simplifying t = 1.540569415Subproblem 2
t + -0.75 = -0.790569415 Simplifying t + -0.75 = -0.790569415 Reorder the terms: -0.75 + t = -0.790569415 Solving -0.75 + t = -0.790569415 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = -0.790569415 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -0.790569415 + 0.75 t = -0.790569415 + 0.75 Combine like terms: -0.790569415 + 0.75 = -0.040569415 t = -0.040569415 Simplifying t = -0.040569415Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.540569415, -0.040569415}
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