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Simplifying -16t2 + 69t + 3 = 0 Reorder the terms: 3 + 69t + -16t2 = 0 Solving 3 + 69t + -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'. -0.1875 + -4.3125t + t2 = 0 Move the constant term to the right: Add '0.1875' to each side of the equation. -0.1875 + -4.3125t + 0.1875 + t2 = 0 + 0.1875 Reorder the terms: -0.1875 + 0.1875 + -4.3125t + t2 = 0 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + -4.3125t + t2 = 0 + 0.1875 -4.3125t + t2 = 0 + 0.1875 Combine like terms: 0 + 0.1875 = 0.1875 -4.3125t + t2 = 0.1875 The t term is -4.3125t. Take half its coefficient (-2.15625). Square it (4.649414063) and add it to both sides. Add '4.649414063' to each side of the equation. -4.3125t + 4.649414063 + t2 = 0.1875 + 4.649414063 Reorder the terms: 4.649414063 + -4.3125t + t2 = 0.1875 + 4.649414063 Combine like terms: 0.1875 + 4.649414063 = 4.836914063 4.649414063 + -4.3125t + t2 = 4.836914063 Factor a perfect square on the left side: (t + -2.15625)(t + -2.15625) = 4.836914063 Calculate the square root of the right side: 2.199298539 Break this problem into two subproblems by setting (t + -2.15625) equal to 2.199298539 and -2.199298539.Subproblem 1
t + -2.15625 = 2.199298539 Simplifying t + -2.15625 = 2.199298539 Reorder the terms: -2.15625 + t = 2.199298539 Solving -2.15625 + t = 2.199298539 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.15625' to each side of the equation. -2.15625 + 2.15625 + t = 2.199298539 + 2.15625 Combine like terms: -2.15625 + 2.15625 = 0.00000 0.00000 + t = 2.199298539 + 2.15625 t = 2.199298539 + 2.15625 Combine like terms: 2.199298539 + 2.15625 = 4.355548539 t = 4.355548539 Simplifying t = 4.355548539Subproblem 2
t + -2.15625 = -2.199298539 Simplifying t + -2.15625 = -2.199298539 Reorder the terms: -2.15625 + t = -2.199298539 Solving -2.15625 + t = -2.199298539 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.15625' to each side of the equation. -2.15625 + 2.15625 + t = -2.199298539 + 2.15625 Combine like terms: -2.15625 + 2.15625 = 0.00000 0.00000 + t = -2.199298539 + 2.15625 t = -2.199298539 + 2.15625 Combine like terms: -2.199298539 + 2.15625 = -0.043048539 t = -0.043048539 Simplifying t = -0.043048539Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.355548539, -0.043048539}
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