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Simplifying 0 = -4t2 + t + 9 Reorder the terms: 0 = 9 + t + -4t2 Solving 0 = 9 + t + -4t2 Solving for variable 't'. Combine like terms: 0 + -9 = -9 -9 + -1t + 4t2 = 9 + t + -4t2 + -9 + -1t + 4t2 Reorder the terms: -9 + -1t + 4t2 = 9 + -9 + t + -1t + -4t2 + 4t2 Combine like terms: 9 + -9 = 0 -9 + -1t + 4t2 = 0 + t + -1t + -4t2 + 4t2 -9 + -1t + 4t2 = t + -1t + -4t2 + 4t2 Combine like terms: t + -1t = 0 -9 + -1t + 4t2 = 0 + -4t2 + 4t2 -9 + -1t + 4t2 = -4t2 + 4t2 Combine like terms: -4t2 + 4t2 = 0 -9 + -1t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -2.25 + -0.25t + t2 = 0 Move the constant term to the right: Add '2.25' to each side of the equation. -2.25 + -0.25t + 2.25 + t2 = 0 + 2.25 Reorder the terms: -2.25 + 2.25 + -0.25t + t2 = 0 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + -0.25t + t2 = 0 + 2.25 -0.25t + t2 = 0 + 2.25 Combine like terms: 0 + 2.25 = 2.25 -0.25t + t2 = 2.25 The t term is -0.25t. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25t + 0.015625 + t2 = 2.25 + 0.015625 Reorder the terms: 0.015625 + -0.25t + t2 = 2.25 + 0.015625 Combine like terms: 2.25 + 0.015625 = 2.265625 0.015625 + -0.25t + t2 = 2.265625 Factor a perfect square on the left side: (t + -0.125)(t + -0.125) = 2.265625 Calculate the square root of the right side: 1.505199322 Break this problem into two subproblems by setting (t + -0.125) equal to 1.505199322 and -1.505199322.Subproblem 1
t + -0.125 = 1.505199322 Simplifying t + -0.125 = 1.505199322 Reorder the terms: -0.125 + t = 1.505199322 Solving -0.125 + t = 1.505199322 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + t = 1.505199322 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + t = 1.505199322 + 0.125 t = 1.505199322 + 0.125 Combine like terms: 1.505199322 + 0.125 = 1.630199322 t = 1.630199322 Simplifying t = 1.630199322Subproblem 2
t + -0.125 = -1.505199322 Simplifying t + -0.125 = -1.505199322 Reorder the terms: -0.125 + t = -1.505199322 Solving -0.125 + t = -1.505199322 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + t = -1.505199322 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + t = -1.505199322 + 0.125 t = -1.505199322 + 0.125 Combine like terms: -1.505199322 + 0.125 = -1.380199322 t = -1.380199322 Simplifying t = -1.380199322Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.630199322, -1.380199322}
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