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Simplifying 0 = -16t2 + 15t + 12 Reorder the terms: 0 = 12 + 15t + -16t2 Solving 0 = 12 + 15t + -16t2 Solving for variable 't'. Combine like terms: 0 + -12 = -12 -12 + -15t + 16t2 = 12 + 15t + -16t2 + -12 + -15t + 16t2 Reorder the terms: -12 + -15t + 16t2 = 12 + -12 + 15t + -15t + -16t2 + 16t2 Combine like terms: 12 + -12 = 0 -12 + -15t + 16t2 = 0 + 15t + -15t + -16t2 + 16t2 -12 + -15t + 16t2 = 15t + -15t + -16t2 + 16t2 Combine like terms: 15t + -15t = 0 -12 + -15t + 16t2 = 0 + -16t2 + 16t2 -12 + -15t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -12 + -15t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.75 + -0.9375t + t2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -0.9375t + 0.75 + t2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -0.9375t + t2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -0.9375t + t2 = 0 + 0.75 -0.9375t + t2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -0.9375t + t2 = 0.75 The t term is -0.9375t. Take half its coefficient (-0.46875). Square it (0.2197265625) and add it to both sides. Add '0.2197265625' to each side of the equation. -0.9375t + 0.2197265625 + t2 = 0.75 + 0.2197265625 Reorder the terms: 0.2197265625 + -0.9375t + t2 = 0.75 + 0.2197265625 Combine like terms: 0.75 + 0.2197265625 = 0.9697265625 0.2197265625 + -0.9375t + t2 = 0.9697265625 Factor a perfect square on the left side: (t + -0.46875)(t + -0.46875) = 0.9697265625 Calculate the square root of the right side: 0.984746954 Break this problem into two subproblems by setting (t + -0.46875) equal to 0.984746954 and -0.984746954.Subproblem 1
t + -0.46875 = 0.984746954 Simplifying t + -0.46875 = 0.984746954 Reorder the terms: -0.46875 + t = 0.984746954 Solving -0.46875 + t = 0.984746954 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.46875' to each side of the equation. -0.46875 + 0.46875 + t = 0.984746954 + 0.46875 Combine like terms: -0.46875 + 0.46875 = 0.00000 0.00000 + t = 0.984746954 + 0.46875 t = 0.984746954 + 0.46875 Combine like terms: 0.984746954 + 0.46875 = 1.453496954 t = 1.453496954 Simplifying t = 1.453496954Subproblem 2
t + -0.46875 = -0.984746954 Simplifying t + -0.46875 = -0.984746954 Reorder the terms: -0.46875 + t = -0.984746954 Solving -0.46875 + t = -0.984746954 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.46875' to each side of the equation. -0.46875 + 0.46875 + t = -0.984746954 + 0.46875 Combine like terms: -0.46875 + 0.46875 = 0.00000 0.00000 + t = -0.984746954 + 0.46875 t = -0.984746954 + 0.46875 Combine like terms: -0.984746954 + 0.46875 = -0.515996954 t = -0.515996954 Simplifying t = -0.515996954Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.453496954, -0.515996954}
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