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Simplifying -16t2 + 32t + 48 = t Reorder the terms: 48 + 32t + -16t2 = t Solving 48 + 32t + -16t2 = t Solving for variable 't'. Reorder the terms: 48 + 32t + -1t + -16t2 = t + -1t Combine like terms: 32t + -1t = 31t 48 + 31t + -16t2 = t + -1t Combine like terms: t + -1t = 0 48 + 31t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -3 + -1.9375t + t2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + -1.9375t + 3 + t2 = 0 + 3 Reorder the terms: -3 + 3 + -1.9375t + t2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + -1.9375t + t2 = 0 + 3 -1.9375t + t2 = 0 + 3 Combine like terms: 0 + 3 = 3 -1.9375t + t2 = 3 The t term is -1.9375t. Take half its coefficient (-0.96875). Square it (0.9384765625) and add it to both sides. Add '0.9384765625' to each side of the equation. -1.9375t + 0.9384765625 + t2 = 3 + 0.9384765625 Reorder the terms: 0.9384765625 + -1.9375t + t2 = 3 + 0.9384765625 Combine like terms: 3 + 0.9384765625 = 3.9384765625 0.9384765625 + -1.9375t + t2 = 3.9384765625 Factor a perfect square on the left side: (t + -0.96875)(t + -0.96875) = 3.9384765625 Calculate the square root of the right side: 1.984559539 Break this problem into two subproblems by setting (t + -0.96875) equal to 1.984559539 and -1.984559539.Subproblem 1
t + -0.96875 = 1.984559539 Simplifying t + -0.96875 = 1.984559539 Reorder the terms: -0.96875 + t = 1.984559539 Solving -0.96875 + t = 1.984559539 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.96875' to each side of the equation. -0.96875 + 0.96875 + t = 1.984559539 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = 1.984559539 + 0.96875 t = 1.984559539 + 0.96875 Combine like terms: 1.984559539 + 0.96875 = 2.953309539 t = 2.953309539 Simplifying t = 2.953309539Subproblem 2
t + -0.96875 = -1.984559539 Simplifying t + -0.96875 = -1.984559539 Reorder the terms: -0.96875 + t = -1.984559539 Solving -0.96875 + t = -1.984559539 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.96875' to each side of the equation. -0.96875 + 0.96875 + t = -1.984559539 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = -1.984559539 + 0.96875 t = -1.984559539 + 0.96875 Combine like terms: -1.984559539 + 0.96875 = -1.015809539 t = -1.015809539 Simplifying t = -1.015809539Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.953309539, -1.015809539}
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