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Simplifying 16t2 + 64t + 8 = t Reorder the terms: 8 + 64t + 16t2 = t Solving 8 + 64t + 16t2 = t Solving for variable 't'. Reorder the terms: 8 + 64t + -1t + 16t2 = t + -1t Combine like terms: 64t + -1t = 63t 8 + 63t + 16t2 = t + -1t Combine like terms: t + -1t = 0 8 + 63t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.5 + 3.9375t + t2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 3.9375t + -0.5 + t2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 3.9375t + t2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 3.9375t + t2 = 0 + -0.5 3.9375t + t2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 3.9375t + t2 = -0.5 The t term is 3.9375t. Take half its coefficient (1.96875). Square it (3.875976563) and add it to both sides. Add '3.875976563' to each side of the equation. 3.9375t + 3.875976563 + t2 = -0.5 + 3.875976563 Reorder the terms: 3.875976563 + 3.9375t + t2 = -0.5 + 3.875976563 Combine like terms: -0.5 + 3.875976563 = 3.375976563 3.875976563 + 3.9375t + t2 = 3.375976563 Factor a perfect square on the left side: (t + 1.96875)(t + 1.96875) = 3.375976563 Calculate the square root of the right side: 1.837383075 Break this problem into two subproblems by setting (t + 1.96875) equal to 1.837383075 and -1.837383075.Subproblem 1
t + 1.96875 = 1.837383075 Simplifying t + 1.96875 = 1.837383075 Reorder the terms: 1.96875 + t = 1.837383075 Solving 1.96875 + t = 1.837383075 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.96875' to each side of the equation. 1.96875 + -1.96875 + t = 1.837383075 + -1.96875 Combine like terms: 1.96875 + -1.96875 = 0.00000 0.00000 + t = 1.837383075 + -1.96875 t = 1.837383075 + -1.96875 Combine like terms: 1.837383075 + -1.96875 = -0.131366925 t = -0.131366925 Simplifying t = -0.131366925Subproblem 2
t + 1.96875 = -1.837383075 Simplifying t + 1.96875 = -1.837383075 Reorder the terms: 1.96875 + t = -1.837383075 Solving 1.96875 + t = -1.837383075 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.96875' to each side of the equation. 1.96875 + -1.96875 + t = -1.837383075 + -1.96875 Combine like terms: 1.96875 + -1.96875 = 0.00000 0.00000 + t = -1.837383075 + -1.96875 t = -1.837383075 + -1.96875 Combine like terms: -1.837383075 + -1.96875 = -3.806133075 t = -3.806133075 Simplifying t = -3.806133075Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.131366925, -3.806133075}
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