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Simplifying -16t2 + 32t + -3 = 10 Reorder the terms: -3 + 32t + -16t2 = 10 Solving -3 + 32t + -16t2 = 10 Solving for variable 't'. Reorder the terms: -3 + -10 + 32t + -16t2 = 10 + -10 Combine like terms: -3 + -10 = -13 -13 + 32t + -16t2 = 10 + -10 Combine like terms: 10 + -10 = 0 -13 + 32t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 0.8125 + -2t + t2 = 0 Move the constant term to the right: Add '-0.8125' to each side of the equation. 0.8125 + -2t + -0.8125 + t2 = 0 + -0.8125 Reorder the terms: 0.8125 + -0.8125 + -2t + t2 = 0 + -0.8125 Combine like terms: 0.8125 + -0.8125 = 0.0000 0.0000 + -2t + t2 = 0 + -0.8125 -2t + t2 = 0 + -0.8125 Combine like terms: 0 + -0.8125 = -0.8125 -2t + t2 = -0.8125 The t term is -2t. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2t + 1 + t2 = -0.8125 + 1 Reorder the terms: 1 + -2t + t2 = -0.8125 + 1 Combine like terms: -0.8125 + 1 = 0.1875 1 + -2t + t2 = 0.1875 Factor a perfect square on the left side: (t + -1)(t + -1) = 0.1875 Calculate the square root of the right side: 0.433012702 Break this problem into two subproblems by setting (t + -1) equal to 0.433012702 and -0.433012702.Subproblem 1
t + -1 = 0.433012702 Simplifying t + -1 = 0.433012702 Reorder the terms: -1 + t = 0.433012702 Solving -1 + t = 0.433012702 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = 0.433012702 + 1 Combine like terms: -1 + 1 = 0 0 + t = 0.433012702 + 1 t = 0.433012702 + 1 Combine like terms: 0.433012702 + 1 = 1.433012702 t = 1.433012702 Simplifying t = 1.433012702Subproblem 2
t + -1 = -0.433012702 Simplifying t + -1 = -0.433012702 Reorder the terms: -1 + t = -0.433012702 Solving -1 + t = -0.433012702 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = -0.433012702 + 1 Combine like terms: -1 + 1 = 0 0 + t = -0.433012702 + 1 t = -0.433012702 + 1 Combine like terms: -0.433012702 + 1 = 0.566987298 t = 0.566987298 Simplifying t = 0.566987298Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.433012702, 0.566987298}
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