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Simplifying 5 + 29t + -16t2 = 12 Solving 5 + 29t + -16t2 = 12 Solving for variable 't'. Reorder the terms: 5 + -12 + 29t + -16t2 = 12 + -12 Combine like terms: 5 + -12 = -7 -7 + 29t + -16t2 = 12 + -12 Combine like terms: 12 + -12 = 0 -7 + 29t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 0.4375 + -1.8125t + t2 = 0 Move the constant term to the right: Add '-0.4375' to each side of the equation. 0.4375 + -1.8125t + -0.4375 + t2 = 0 + -0.4375 Reorder the terms: 0.4375 + -0.4375 + -1.8125t + t2 = 0 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + -1.8125t + t2 = 0 + -0.4375 -1.8125t + t2 = 0 + -0.4375 Combine like terms: 0 + -0.4375 = -0.4375 -1.8125t + t2 = -0.4375 The t term is -1.8125t. Take half its coefficient (-0.90625). Square it (0.8212890625) and add it to both sides. Add '0.8212890625' to each side of the equation. -1.8125t + 0.8212890625 + t2 = -0.4375 + 0.8212890625 Reorder the terms: 0.8212890625 + -1.8125t + t2 = -0.4375 + 0.8212890625 Combine like terms: -0.4375 + 0.8212890625 = 0.3837890625 0.8212890625 + -1.8125t + t2 = 0.3837890625 Factor a perfect square on the left side: (t + -0.90625)(t + -0.90625) = 0.3837890625 Calculate the square root of the right side: 0.619507113 Break this problem into two subproblems by setting (t + -0.90625) equal to 0.619507113 and -0.619507113.Subproblem 1
t + -0.90625 = 0.619507113 Simplifying t + -0.90625 = 0.619507113 Reorder the terms: -0.90625 + t = 0.619507113 Solving -0.90625 + t = 0.619507113 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.90625' to each side of the equation. -0.90625 + 0.90625 + t = 0.619507113 + 0.90625 Combine like terms: -0.90625 + 0.90625 = 0.00000 0.00000 + t = 0.619507113 + 0.90625 t = 0.619507113 + 0.90625 Combine like terms: 0.619507113 + 0.90625 = 1.525757113 t = 1.525757113 Simplifying t = 1.525757113Subproblem 2
t + -0.90625 = -0.619507113 Simplifying t + -0.90625 = -0.619507113 Reorder the terms: -0.90625 + t = -0.619507113 Solving -0.90625 + t = -0.619507113 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.90625' to each side of the equation. -0.90625 + 0.90625 + t = -0.619507113 + 0.90625 Combine like terms: -0.90625 + 0.90625 = 0.00000 0.00000 + t = -0.619507113 + 0.90625 t = -0.619507113 + 0.90625 Combine like terms: -0.619507113 + 0.90625 = 0.286742887 t = 0.286742887 Simplifying t = 0.286742887Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.525757113, 0.286742887}
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