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Simplifying 0 = 7 + 24t + 16t2 Solving 0 = 7 + 24t + 16t2 Solving for variable 't'. Combine like terms: 0 + -7 = -7 -7 + -24t + -16t2 = 7 + 24t + 16t2 + -7 + -24t + -16t2 Reorder the terms: -7 + -24t + -16t2 = 7 + -7 + 24t + -24t + 16t2 + -16t2 Combine like terms: 7 + -7 = 0 -7 + -24t + -16t2 = 0 + 24t + -24t + 16t2 + -16t2 -7 + -24t + -16t2 = 24t + -24t + 16t2 + -16t2 Combine like terms: 24t + -24t = 0 -7 + -24t + -16t2 = 0 + 16t2 + -16t2 -7 + -24t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -7 + -24t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(7 + 24t + 16t2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(7 + 24t + 16t2)' equal to zero and attempt to solve: Simplifying 7 + 24t + 16t2 = 0 Solving 7 + 24t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.4375 + 1.5t + t2 = 0 Move the constant term to the right: Add '-0.4375' to each side of the equation. 0.4375 + 1.5t + -0.4375 + t2 = 0 + -0.4375 Reorder the terms: 0.4375 + -0.4375 + 1.5t + t2 = 0 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + 1.5t + t2 = 0 + -0.4375 1.5t + t2 = 0 + -0.4375 Combine like terms: 0 + -0.4375 = -0.4375 1.5t + t2 = -0.4375 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 = -0.4375 + 0.5625 Reorder the terms: 0.5625 + 1.5t + t2 = -0.4375 + 0.5625 Combine like terms: -0.4375 + 0.5625 = 0.125 0.5625 + 1.5t + t2 = 0.125 Factor a perfect square on the left side: (t + 0.75)(t + 0.75) = 0.125 Calculate the square root of the right side: 0.353553391 Break this problem into two subproblems by setting (t + 0.75) equal to 0.353553391 and -0.353553391.Subproblem 1
t + 0.75 = 0.353553391 Simplifying t + 0.75 = 0.353553391 Reorder the terms: 0.75 + t = 0.353553391 Solving 0.75 + t = 0.353553391 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 = 0.353553391 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + t = 0.353553391 + -0.75 t = 0.353553391 + -0.75 Combine like terms: 0.353553391 + -0.75 = -0.396446609 t = -0.396446609 Simplifying t = -0.396446609Subproblem 2
t + 0.75 = -0.353553391 Simplifying t + 0.75 = -0.353553391 Reorder the terms: 0.75 + t = -0.353553391 Solving 0.75 + t = -0.353553391 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 = -0.353553391 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + t = -0.353553391 + -0.75 t = -0.353553391 + -0.75 Combine like terms: -0.353553391 + -0.75 = -1.103553391 t = -1.103553391 Simplifying t = -1.103553391Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.396446609, -1.103553391}Solution
t = {-0.396446609, -1.103553391}
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