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Simplifying -16t2 + 40t + 6 = 3 Reorder the terms: 6 + 40t + -16t2 = 3 Solving 6 + 40t + -16t2 = 3 Solving for variable 't'. Reorder the terms: 6 + -3 + 40t + -16t2 = 3 + -3 Combine like terms: 6 + -3 = 3 3 + 40t + -16t2 = 3 + -3 Combine like terms: 3 + -3 = 0 3 + 40t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.1875 + -2.5t + t2 = 0 Move the constant term to the right: Add '0.1875' to each side of the equation. -0.1875 + -2.5t + 0.1875 + t2 = 0 + 0.1875 Reorder the terms: -0.1875 + 0.1875 + -2.5t + t2 = 0 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + -2.5t + t2 = 0 + 0.1875 -2.5t + t2 = 0 + 0.1875 Combine like terms: 0 + 0.1875 = 0.1875 -2.5t + t2 = 0.1875 The t term is -2.5t. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5t + 1.5625 + t2 = 0.1875 + 1.5625 Reorder the terms: 1.5625 + -2.5t + t2 = 0.1875 + 1.5625 Combine like terms: 0.1875 + 1.5625 = 1.75 1.5625 + -2.5t + t2 = 1.75 Factor a perfect square on the left side: (t + -1.25)(t + -1.25) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (t + -1.25) equal to 1.322875656 and -1.322875656.Subproblem 1
t + -1.25 = 1.322875656 Simplifying t + -1.25 = 1.322875656 Reorder the terms: -1.25 + t = 1.322875656 Solving -1.25 + t = 1.322875656 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = 1.322875656 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = 1.322875656 + 1.25 t = 1.322875656 + 1.25 Combine like terms: 1.322875656 + 1.25 = 2.572875656 t = 2.572875656 Simplifying t = 2.572875656Subproblem 2
t + -1.25 = -1.322875656 Simplifying t + -1.25 = -1.322875656 Reorder the terms: -1.25 + t = -1.322875656 Solving -1.25 + t = -1.322875656 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = -1.322875656 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = -1.322875656 + 1.25 t = -1.322875656 + 1.25 Combine like terms: -1.322875656 + 1.25 = -0.072875656 t = -0.072875656 Simplifying t = -0.072875656Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.572875656, -0.072875656}
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