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Simplifying 7 = 3 + 24t + -16t2 Solving 7 = 3 + 24t + -16t2 Solving for variable 't'. Combine like terms: 7 + -3 = 4 4 + -24t + 16t2 = 3 + 24t + -16t2 + -3 + -24t + 16t2 Reorder the terms: 4 + -24t + 16t2 = 3 + -3 + 24t + -24t + -16t2 + 16t2 Combine like terms: 3 + -3 = 0 4 + -24t + 16t2 = 0 + 24t + -24t + -16t2 + 16t2 4 + -24t + 16t2 = 24t + -24t + -16t2 + 16t2 Combine like terms: 24t + -24t = 0 4 + -24t + 16t2 = 0 + -16t2 + 16t2 4 + -24t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 4 + -24t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(1 + -6t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(1 + -6t + 4t2)' equal to zero and attempt to solve: Simplifying 1 + -6t + 4t2 = 0 Solving 1 + -6t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.25 + -1.5t + t2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + -1.5t + -0.25 + t2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + -1.5t + t2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + -1.5t + t2 = 0 + -0.25 -1.5t + t2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 -1.5t + t2 = -0.25 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 = -0.25 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = -0.25 + 0.5625 Combine like terms: -0.25 + 0.5625 = 0.3125 0.5625 + -1.5t + t2 = 0.3125 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 0.3125 Calculate the square root of the right side: 0.559016994 Break this problem into two subproblems by setting (t + -0.75) equal to 0.559016994 and -0.559016994.Subproblem 1
t + -0.75 = 0.559016994 Simplifying t + -0.75 = 0.559016994 Reorder the terms: -0.75 + t = 0.559016994 Solving -0.75 + t = 0.559016994 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 = 0.559016994 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 0.559016994 + 0.75 t = 0.559016994 + 0.75 Combine like terms: 0.559016994 + 0.75 = 1.309016994 t = 1.309016994 Simplifying t = 1.309016994Subproblem 2
t + -0.75 = -0.559016994 Simplifying t + -0.75 = -0.559016994 Reorder the terms: -0.75 + t = -0.559016994 Solving -0.75 + t = -0.559016994 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 = -0.559016994 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -0.559016994 + 0.75 t = -0.559016994 + 0.75 Combine like terms: -0.559016994 + 0.75 = 0.190983006 t = 0.190983006 Simplifying t = 0.190983006Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.309016994, 0.190983006}Solution
t = {1.309016994, 0.190983006}
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