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Simplifying -4t2 + 7t = -32 Reorder the terms: 7t + -4t2 = -32 Solving 7t + -4t2 = -32 Solving for variable 't'. Reorder the terms: 32 + 7t + -4t2 = -32 + 32 Combine like terms: -32 + 32 = 0 32 + 7t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -8 + -1.75t + t2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + -1.75t + 8 + t2 = 0 + 8 Reorder the terms: -8 + 8 + -1.75t + t2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + -1.75t + t2 = 0 + 8 -1.75t + t2 = 0 + 8 Combine like terms: 0 + 8 = 8 -1.75t + t2 = 8 The t term is -1.75t. Take half its coefficient (-0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. -1.75t + 0.765625 + t2 = 8 + 0.765625 Reorder the terms: 0.765625 + -1.75t + t2 = 8 + 0.765625 Combine like terms: 8 + 0.765625 = 8.765625 0.765625 + -1.75t + t2 = 8.765625 Factor a perfect square on the left side: (t + -0.875)(t + -0.875) = 8.765625 Calculate the square root of the right side: 2.960679821 Break this problem into two subproblems by setting (t + -0.875) equal to 2.960679821 and -2.960679821.Subproblem 1
t + -0.875 = 2.960679821 Simplifying t + -0.875 = 2.960679821 Reorder the terms: -0.875 + t = 2.960679821 Solving -0.875 + t = 2.960679821 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + t = 2.960679821 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + t = 2.960679821 + 0.875 t = 2.960679821 + 0.875 Combine like terms: 2.960679821 + 0.875 = 3.835679821 t = 3.835679821 Simplifying t = 3.835679821Subproblem 2
t + -0.875 = -2.960679821 Simplifying t + -0.875 = -2.960679821 Reorder the terms: -0.875 + t = -2.960679821 Solving -0.875 + t = -2.960679821 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + t = -2.960679821 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + t = -2.960679821 + 0.875 t = -2.960679821 + 0.875 Combine like terms: -2.960679821 + 0.875 = -2.085679821 t = -2.085679821 Simplifying t = -2.085679821Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.835679821, -2.085679821}
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