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Simplifying 14 = -16t2 + 25t + 6 Reorder the terms: 14 = 6 + 25t + -16t2 Solving 14 = 6 + 25t + -16t2 Solving for variable 't'. Combine like terms: 14 + -6 = 8 8 + -25t + 16t2 = 6 + 25t + -16t2 + -6 + -25t + 16t2 Reorder the terms: 8 + -25t + 16t2 = 6 + -6 + 25t + -25t + -16t2 + 16t2 Combine like terms: 6 + -6 = 0 8 + -25t + 16t2 = 0 + 25t + -25t + -16t2 + 16t2 8 + -25t + 16t2 = 25t + -25t + -16t2 + 16t2 Combine like terms: 25t + -25t = 0 8 + -25t + 16t2 = 0 + -16t2 + 16t2 8 + -25t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 8 + -25t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.5 + -1.5625t + t2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -1.5625t + -0.5 + t2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -1.5625t + t2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -1.5625t + t2 = 0 + -0.5 -1.5625t + t2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -1.5625t + t2 = -0.5 The t term is -1.5625t. Take half its coefficient (-0.78125). Square it (0.6103515625) and add it to both sides. Add '0.6103515625' to each side of the equation. -1.5625t + 0.6103515625 + t2 = -0.5 + 0.6103515625 Reorder the terms: 0.6103515625 + -1.5625t + t2 = -0.5 + 0.6103515625 Combine like terms: -0.5 + 0.6103515625 = 0.1103515625 0.6103515625 + -1.5625t + t2 = 0.1103515625 Factor a perfect square on the left side: (t + -0.78125)(t + -0.78125) = 0.1103515625 Calculate the square root of the right side: 0.332192057 Break this problem into two subproblems by setting (t + -0.78125) equal to 0.332192057 and -0.332192057.Subproblem 1
t + -0.78125 = 0.332192057 Simplifying t + -0.78125 = 0.332192057 Reorder the terms: -0.78125 + t = 0.332192057 Solving -0.78125 + t = 0.332192057 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.78125' to each side of the equation. -0.78125 + 0.78125 + t = 0.332192057 + 0.78125 Combine like terms: -0.78125 + 0.78125 = 0.00000 0.00000 + t = 0.332192057 + 0.78125 t = 0.332192057 + 0.78125 Combine like terms: 0.332192057 + 0.78125 = 1.113442057 t = 1.113442057 Simplifying t = 1.113442057Subproblem 2
t + -0.78125 = -0.332192057 Simplifying t + -0.78125 = -0.332192057 Reorder the terms: -0.78125 + t = -0.332192057 Solving -0.78125 + t = -0.332192057 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.78125' to each side of the equation. -0.78125 + 0.78125 + t = -0.332192057 + 0.78125 Combine like terms: -0.78125 + 0.78125 = 0.00000 0.00000 + t = -0.332192057 + 0.78125 t = -0.332192057 + 0.78125 Combine like terms: -0.332192057 + 0.78125 = 0.449057943 t = 0.449057943 Simplifying t = 0.449057943Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.113442057, 0.449057943}
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