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Simplifying 9 = 105 + 9t + -16t2 Solving 9 = 105 + 9t + -16t2 Solving for variable 't'. Combine like terms: 9 + -105 = -96 -96 + -9t + 16t2 = 105 + 9t + -16t2 + -105 + -9t + 16t2 Reorder the terms: -96 + -9t + 16t2 = 105 + -105 + 9t + -9t + -16t2 + 16t2 Combine like terms: 105 + -105 = 0 -96 + -9t + 16t2 = 0 + 9t + -9t + -16t2 + 16t2 -96 + -9t + 16t2 = 9t + -9t + -16t2 + 16t2 Combine like terms: 9t + -9t = 0 -96 + -9t + 16t2 = 0 + -16t2 + 16t2 -96 + -9t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -96 + -9t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -6 + -0.5625t + t2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + -0.5625t + 6 + t2 = 0 + 6 Reorder the terms: -6 + 6 + -0.5625t + t2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -0.5625t + t2 = 0 + 6 -0.5625t + t2 = 0 + 6 Combine like terms: 0 + 6 = 6 -0.5625t + t2 = 6 The t term is -0.5625t. Take half its coefficient (-0.28125). Square it (0.0791015625) and add it to both sides. Add '0.0791015625' to each side of the equation. -0.5625t + 0.0791015625 + t2 = 6 + 0.0791015625 Reorder the terms: 0.0791015625 + -0.5625t + t2 = 6 + 0.0791015625 Combine like terms: 6 + 0.0791015625 = 6.0791015625 0.0791015625 + -0.5625t + t2 = 6.0791015625 Factor a perfect square on the left side: (t + -0.28125)(t + -0.28125) = 6.0791015625 Calculate the square root of the right side: 2.465583412 Break this problem into two subproblems by setting (t + -0.28125) equal to 2.465583412 and -2.465583412.Subproblem 1
t + -0.28125 = 2.465583412 Simplifying t + -0.28125 = 2.465583412 Reorder the terms: -0.28125 + t = 2.465583412 Solving -0.28125 + t = 2.465583412 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.28125' to each side of the equation. -0.28125 + 0.28125 + t = 2.465583412 + 0.28125 Combine like terms: -0.28125 + 0.28125 = 0.00000 0.00000 + t = 2.465583412 + 0.28125 t = 2.465583412 + 0.28125 Combine like terms: 2.465583412 + 0.28125 = 2.746833412 t = 2.746833412 Simplifying t = 2.746833412Subproblem 2
t + -0.28125 = -2.465583412 Simplifying t + -0.28125 = -2.465583412 Reorder the terms: -0.28125 + t = -2.465583412 Solving -0.28125 + t = -2.465583412 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.28125' to each side of the equation. -0.28125 + 0.28125 + t = -2.465583412 + 0.28125 Combine like terms: -0.28125 + 0.28125 = 0.00000 0.00000 + t = -2.465583412 + 0.28125 t = -2.465583412 + 0.28125 Combine like terms: -2.465583412 + 0.28125 = -2.184333412 t = -2.184333412 Simplifying t = -2.184333412Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.746833412, -2.184333412}
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