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Simplifying -16t2 + 20t + 15 = 5 Reorder the terms: 15 + 20t + -16t2 = 5 Solving 15 + 20t + -16t2 = 5 Solving for variable 't'. Reorder the terms: 15 + -5 + 20t + -16t2 = 5 + -5 Combine like terms: 15 + -5 = 10 10 + 20t + -16t2 = 5 + -5 Combine like terms: 5 + -5 = 0 10 + 20t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(5 + 10t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(5 + 10t + -8t2)' equal to zero and attempt to solve: Simplifying 5 + 10t + -8t2 = 0 Solving 5 + 10t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.625 + -1.25t + t2 = 0 Move the constant term to the right: Add '0.625' to each side of the equation. -0.625 + -1.25t + 0.625 + t2 = 0 + 0.625 Reorder the terms: -0.625 + 0.625 + -1.25t + t2 = 0 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + -1.25t + t2 = 0 + 0.625 -1.25t + t2 = 0 + 0.625 Combine like terms: 0 + 0.625 = 0.625 -1.25t + t2 = 0.625 The t term is -1.25t. Take half its coefficient (-0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. -1.25t + 0.390625 + t2 = 0.625 + 0.390625 Reorder the terms: 0.390625 + -1.25t + t2 = 0.625 + 0.390625 Combine like terms: 0.625 + 0.390625 = 1.015625 0.390625 + -1.25t + t2 = 1.015625 Factor a perfect square on the left side: (t + -0.625)(t + -0.625) = 1.015625 Calculate the square root of the right side: 1.007782219 Break this problem into two subproblems by setting (t + -0.625) equal to 1.007782219 and -1.007782219.Subproblem 1
t + -0.625 = 1.007782219 Simplifying t + -0.625 = 1.007782219 Reorder the terms: -0.625 + t = 1.007782219 Solving -0.625 + t = 1.007782219 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + t = 1.007782219 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + t = 1.007782219 + 0.625 t = 1.007782219 + 0.625 Combine like terms: 1.007782219 + 0.625 = 1.632782219 t = 1.632782219 Simplifying t = 1.632782219Subproblem 2
t + -0.625 = -1.007782219 Simplifying t + -0.625 = -1.007782219 Reorder the terms: -0.625 + t = -1.007782219 Solving -0.625 + t = -1.007782219 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + t = -1.007782219 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + t = -1.007782219 + 0.625 t = -1.007782219 + 0.625 Combine like terms: -1.007782219 + 0.625 = -0.382782219 t = -0.382782219 Simplifying t = -0.382782219Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.632782219, -0.382782219}Solution
t = {1.632782219, -0.382782219}
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