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Simplifying -162t2 + 112t + 128 = 0 Reorder the terms: 128 + 112t + -162t2 = 0 Solving 128 + 112t + -162t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(64 + 56t + -81t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(64 + 56t + -81t2)' equal to zero and attempt to solve: Simplifying 64 + 56t + -81t2 = 0 Solving 64 + 56t + -81t2 = 0 Begin completing the square. Divide all terms by -81 the coefficient of the squared term: Divide each side by '-81'. -0.7901234568 + -0.6913580247t + t2 = 0 Move the constant term to the right: Add '0.7901234568' to each side of the equation. -0.7901234568 + -0.6913580247t + 0.7901234568 + t2 = 0 + 0.7901234568 Reorder the terms: -0.7901234568 + 0.7901234568 + -0.6913580247t + t2 = 0 + 0.7901234568 Combine like terms: -0.7901234568 + 0.7901234568 = 0.0000000000 0.0000000000 + -0.6913580247t + t2 = 0 + 0.7901234568 -0.6913580247t + t2 = 0 + 0.7901234568 Combine like terms: 0 + 0.7901234568 = 0.7901234568 -0.6913580247t + t2 = 0.7901234568 The t term is -0.6913580247t. Take half its coefficient (-0.3456790124). Square it (0.1194939796) and add it to both sides. Add '0.1194939796' to each side of the equation. -0.6913580247t + 0.1194939796 + t2 = 0.7901234568 + 0.1194939796 Reorder the terms: 0.1194939796 + -0.6913580247t + t2 = 0.7901234568 + 0.1194939796 Combine like terms: 0.7901234568 + 0.1194939796 = 0.9096174364 0.1194939796 + -0.6913580247t + t2 = 0.9096174364 Factor a perfect square on the left side: (t + -0.3456790124)(t + -0.3456790124) = 0.9096174364 Calculate the square root of the right side: 0.953738663 Break this problem into two subproblems by setting (t + -0.3456790124) equal to 0.953738663 and -0.953738663.Subproblem 1
t + -0.3456790124 = 0.953738663 Simplifying t + -0.3456790124 = 0.953738663 Reorder the terms: -0.3456790124 + t = 0.953738663 Solving -0.3456790124 + t = 0.953738663 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.3456790124' to each side of the equation. -0.3456790124 + 0.3456790124 + t = 0.953738663 + 0.3456790124 Combine like terms: -0.3456790124 + 0.3456790124 = 0.0000000000 0.0000000000 + t = 0.953738663 + 0.3456790124 t = 0.953738663 + 0.3456790124 Combine like terms: 0.953738663 + 0.3456790124 = 1.2994176754 t = 1.2994176754 Simplifying t = 1.2994176754Subproblem 2
t + -0.3456790124 = -0.953738663 Simplifying t + -0.3456790124 = -0.953738663 Reorder the terms: -0.3456790124 + t = -0.953738663 Solving -0.3456790124 + t = -0.953738663 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.3456790124' to each side of the equation. -0.3456790124 + 0.3456790124 + t = -0.953738663 + 0.3456790124 Combine like terms: -0.3456790124 + 0.3456790124 = 0.0000000000 0.0000000000 + t = -0.953738663 + 0.3456790124 t = -0.953738663 + 0.3456790124 Combine like terms: -0.953738663 + 0.3456790124 = -0.6080596506 t = -0.6080596506 Simplifying t = -0.6080596506Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.2994176754, -0.6080596506}Solution
t = {1.2994176754, -0.6080596506}
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