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Simplifying 415 = -16t2 + 164t + 80 Reorder the terms: 415 = 80 + 164t + -16t2 Solving 415 = 80 + 164t + -16t2 Solving for variable 't'. Combine like terms: 415 + -80 = 335 335 + -164t + 16t2 = 80 + 164t + -16t2 + -80 + -164t + 16t2 Reorder the terms: 335 + -164t + 16t2 = 80 + -80 + 164t + -164t + -16t2 + 16t2 Combine like terms: 80 + -80 = 0 335 + -164t + 16t2 = 0 + 164t + -164t + -16t2 + 16t2 335 + -164t + 16t2 = 164t + -164t + -16t2 + 16t2 Combine like terms: 164t + -164t = 0 335 + -164t + 16t2 = 0 + -16t2 + 16t2 335 + -164t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 335 + -164t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 20.9375 + -10.25t + t2 = 0 Move the constant term to the right: Add '-20.9375' to each side of the equation. 20.9375 + -10.25t + -20.9375 + t2 = 0 + -20.9375 Reorder the terms: 20.9375 + -20.9375 + -10.25t + t2 = 0 + -20.9375 Combine like terms: 20.9375 + -20.9375 = 0.0000 0.0000 + -10.25t + t2 = 0 + -20.9375 -10.25t + t2 = 0 + -20.9375 Combine like terms: 0 + -20.9375 = -20.9375 -10.25t + t2 = -20.9375 The t term is -10.25t. Take half its coefficient (-5.125). Square it (26.265625) and add it to both sides. Add '26.265625' to each side of the equation. -10.25t + 26.265625 + t2 = -20.9375 + 26.265625 Reorder the terms: 26.265625 + -10.25t + t2 = -20.9375 + 26.265625 Combine like terms: -20.9375 + 26.265625 = 5.328125 26.265625 + -10.25t + t2 = 5.328125 Factor a perfect square on the left side: (t + -5.125)(t + -5.125) = 5.328125 Calculate the square root of the right side: 2.308273164 Break this problem into two subproblems by setting (t + -5.125) equal to 2.308273164 and -2.308273164.Subproblem 1
t + -5.125 = 2.308273164 Simplifying t + -5.125 = 2.308273164 Reorder the terms: -5.125 + t = 2.308273164 Solving -5.125 + t = 2.308273164 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5.125' to each side of the equation. -5.125 + 5.125 + t = 2.308273164 + 5.125 Combine like terms: -5.125 + 5.125 = 0.000 0.000 + t = 2.308273164 + 5.125 t = 2.308273164 + 5.125 Combine like terms: 2.308273164 + 5.125 = 7.433273164 t = 7.433273164 Simplifying t = 7.433273164Subproblem 2
t + -5.125 = -2.308273164 Simplifying t + -5.125 = -2.308273164 Reorder the terms: -5.125 + t = -2.308273164 Solving -5.125 + t = -2.308273164 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5.125' to each side of the equation. -5.125 + 5.125 + t = -2.308273164 + 5.125 Combine like terms: -5.125 + 5.125 = 0.000 0.000 + t = -2.308273164 + 5.125 t = -2.308273164 + 5.125 Combine like terms: -2.308273164 + 5.125 = 2.816726836 t = 2.816726836 Simplifying t = 2.816726836Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.433273164, 2.816726836}
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