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Simplifying -16t2 + 12t + 52 = 0 Reorder the terms: 52 + 12t + -16t2 = 0 Solving 52 + 12t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '4'. 4(13 + 3t + -4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(13 + 3t + -4t2)' equal to zero and attempt to solve: Simplifying 13 + 3t + -4t2 = 0 Solving 13 + 3t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -3.25 + -0.75t + t2 = 0 Move the constant term to the right: Add '3.25' to each side of the equation. -3.25 + -0.75t + 3.25 + t2 = 0 + 3.25 Reorder the terms: -3.25 + 3.25 + -0.75t + t2 = 0 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + -0.75t + t2 = 0 + 3.25 -0.75t + t2 = 0 + 3.25 Combine like terms: 0 + 3.25 = 3.25 -0.75t + t2 = 3.25 The t term is -0.75t. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75t + 0.140625 + t2 = 3.25 + 0.140625 Reorder the terms: 0.140625 + -0.75t + t2 = 3.25 + 0.140625 Combine like terms: 3.25 + 0.140625 = 3.390625 0.140625 + -0.75t + t2 = 3.390625 Factor a perfect square on the left side: (t + -0.375)(t + -0.375) = 3.390625 Calculate the square root of the right side: 1.841364983 Break this problem into two subproblems by setting (t + -0.375) equal to 1.841364983 and -1.841364983.Subproblem 1
t + -0.375 = 1.841364983 Simplifying t + -0.375 = 1.841364983 Reorder the terms: -0.375 + t = 1.841364983 Solving -0.375 + t = 1.841364983 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + t = 1.841364983 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = 1.841364983 + 0.375 t = 1.841364983 + 0.375 Combine like terms: 1.841364983 + 0.375 = 2.216364983 t = 2.216364983 Simplifying t = 2.216364983Subproblem 2
t + -0.375 = -1.841364983 Simplifying t + -0.375 = -1.841364983 Reorder the terms: -0.375 + t = -1.841364983 Solving -0.375 + t = -1.841364983 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + t = -1.841364983 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = -1.841364983 + 0.375 t = -1.841364983 + 0.375 Combine like terms: -1.841364983 + 0.375 = -1.466364983 t = -1.466364983 Simplifying t = -1.466364983Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.216364983, -1.466364983}Solution
t = {2.216364983, -1.466364983}
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