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Simplifying -16t2 + 46t + 756 = 0 Reorder the terms: 756 + 46t + -16t2 = 0 Solving 756 + 46t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(378 + 23t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(378 + 23t + -8t2)' equal to zero and attempt to solve: Simplifying 378 + 23t + -8t2 = 0 Solving 378 + 23t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -47.25 + -2.875t + t2 = 0 Move the constant term to the right: Add '47.25' to each side of the equation. -47.25 + -2.875t + 47.25 + t2 = 0 + 47.25 Reorder the terms: -47.25 + 47.25 + -2.875t + t2 = 0 + 47.25 Combine like terms: -47.25 + 47.25 = 0.00 0.00 + -2.875t + t2 = 0 + 47.25 -2.875t + t2 = 0 + 47.25 Combine like terms: 0 + 47.25 = 47.25 -2.875t + t2 = 47.25 The t term is -2.875t. Take half its coefficient (-1.4375). Square it (2.06640625) and add it to both sides. Add '2.06640625' to each side of the equation. -2.875t + 2.06640625 + t2 = 47.25 + 2.06640625 Reorder the terms: 2.06640625 + -2.875t + t2 = 47.25 + 2.06640625 Combine like terms: 47.25 + 2.06640625 = 49.31640625 2.06640625 + -2.875t + t2 = 49.31640625 Factor a perfect square on the left side: (t + -1.4375)(t + -1.4375) = 49.31640625 Calculate the square root of the right side: 7.022564079 Break this problem into two subproblems by setting (t + -1.4375) equal to 7.022564079 and -7.022564079.Subproblem 1
t + -1.4375 = 7.022564079 Simplifying t + -1.4375 = 7.022564079 Reorder the terms: -1.4375 + t = 7.022564079 Solving -1.4375 + t = 7.022564079 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.4375' to each side of the equation. -1.4375 + 1.4375 + t = 7.022564079 + 1.4375 Combine like terms: -1.4375 + 1.4375 = 0.0000 0.0000 + t = 7.022564079 + 1.4375 t = 7.022564079 + 1.4375 Combine like terms: 7.022564079 + 1.4375 = 8.460064079 t = 8.460064079 Simplifying t = 8.460064079Subproblem 2
t + -1.4375 = -7.022564079 Simplifying t + -1.4375 = -7.022564079 Reorder the terms: -1.4375 + t = -7.022564079 Solving -1.4375 + t = -7.022564079 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.4375' to each side of the equation. -1.4375 + 1.4375 + t = -7.022564079 + 1.4375 Combine like terms: -1.4375 + 1.4375 = 0.0000 0.0000 + t = -7.022564079 + 1.4375 t = -7.022564079 + 1.4375 Combine like terms: -7.022564079 + 1.4375 = -5.585064079 t = -5.585064079 Simplifying t = -5.585064079Solution
The solution to the problem is based on the solutions from the subproblems. t = {8.460064079, -5.585064079}Solution
t = {8.460064079, -5.585064079}
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