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Simplifying -16t2 + 32t + 128 = t Reorder the terms: 128 + 32t + -16t2 = t Solving 128 + 32t + -16t2 = t Solving for variable 't'. Reorder the terms: 128 + 32t + -1t + -16t2 = t + -1t Combine like terms: 32t + -1t = 31t 128 + 31t + -16t2 = t + -1t Combine like terms: t + -1t = 0 128 + 31t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -8 + -1.9375t + t2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + -1.9375t + 8 + t2 = 0 + 8 Reorder the terms: -8 + 8 + -1.9375t + t2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + -1.9375t + t2 = 0 + 8 -1.9375t + t2 = 0 + 8 Combine like terms: 0 + 8 = 8 -1.9375t + t2 = 8 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 = 8 + 0.9384765625 Reorder the terms: 0.9384765625 + -1.9375t + t2 = 8 + 0.9384765625 Combine like terms: 8 + 0.9384765625 = 8.9384765625 0.9384765625 + -1.9375t + t2 = 8.9384765625 Factor a perfect square on the left side: (t + -0.96875)(t + -0.96875) = 8.9384765625 Calculate the square root of the right side: 2.98972851 Break this problem into two subproblems by setting (t + -0.96875) equal to 2.98972851 and -2.98972851.Subproblem 1
t + -0.96875 = 2.98972851 Simplifying t + -0.96875 = 2.98972851 Reorder the terms: -0.96875 + t = 2.98972851 Solving -0.96875 + t = 2.98972851 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 = 2.98972851 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = 2.98972851 + 0.96875 t = 2.98972851 + 0.96875 Combine like terms: 2.98972851 + 0.96875 = 3.95847851 t = 3.95847851 Simplifying t = 3.95847851Subproblem 2
t + -0.96875 = -2.98972851 Simplifying t + -0.96875 = -2.98972851 Reorder the terms: -0.96875 + t = -2.98972851 Solving -0.96875 + t = -2.98972851 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 = -2.98972851 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = -2.98972851 + 0.96875 t = -2.98972851 + 0.96875 Combine like terms: -2.98972851 + 0.96875 = -2.02097851 t = -2.02097851 Simplifying t = -2.02097851Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.95847851, -2.02097851}
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