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Simplifying -4t(t + -4) + 8t = 9t + -7 Reorder the terms: -4t(-4 + t) + 8t = 9t + -7 (-4 * -4t + t * -4t) + 8t = 9t + -7 (16t + -4t2) + 8t = 9t + -7 Reorder the terms: 16t + 8t + -4t2 = 9t + -7 Combine like terms: 16t + 8t = 24t 24t + -4t2 = 9t + -7 Reorder the terms: 24t + -4t2 = -7 + 9t Solving 24t + -4t2 = -7 + 9t Solving for variable 't'. Reorder the terms: 7 + 24t + -9t + -4t2 = -7 + 9t + 7 + -9t Combine like terms: 24t + -9t = 15t 7 + 15t + -4t2 = -7 + 9t + 7 + -9t Reorder the terms: 7 + 15t + -4t2 = -7 + 7 + 9t + -9t Combine like terms: -7 + 7 = 0 7 + 15t + -4t2 = 0 + 9t + -9t 7 + 15t + -4t2 = 9t + -9t Combine like terms: 9t + -9t = 0 7 + 15t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -1.75 + -3.75t + t2 = 0 Move the constant term to the right: Add '1.75' to each side of the equation. -1.75 + -3.75t + 1.75 + t2 = 0 + 1.75 Reorder the terms: -1.75 + 1.75 + -3.75t + t2 = 0 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + -3.75t + t2 = 0 + 1.75 -3.75t + t2 = 0 + 1.75 Combine like terms: 0 + 1.75 = 1.75 -3.75t + t2 = 1.75 The t term is -3.75t. Take half its coefficient (-1.875). Square it (3.515625) and add it to both sides. Add '3.515625' to each side of the equation. -3.75t + 3.515625 + t2 = 1.75 + 3.515625 Reorder the terms: 3.515625 + -3.75t + t2 = 1.75 + 3.515625 Combine like terms: 1.75 + 3.515625 = 5.265625 3.515625 + -3.75t + t2 = 5.265625 Factor a perfect square on the left side: (t + -1.875)(t + -1.875) = 5.265625 Calculate the square root of the right side: 2.294694969 Break this problem into two subproblems by setting (t + -1.875) equal to 2.294694969 and -2.294694969.Subproblem 1
t + -1.875 = 2.294694969 Simplifying t + -1.875 = 2.294694969 Reorder the terms: -1.875 + t = 2.294694969 Solving -1.875 + t = 2.294694969 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + t = 2.294694969 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + t = 2.294694969 + 1.875 t = 2.294694969 + 1.875 Combine like terms: 2.294694969 + 1.875 = 4.169694969 t = 4.169694969 Simplifying t = 4.169694969Subproblem 2
t + -1.875 = -2.294694969 Simplifying t + -1.875 = -2.294694969 Reorder the terms: -1.875 + t = -2.294694969 Solving -1.875 + t = -2.294694969 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + t = -2.294694969 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + t = -2.294694969 + 1.875 t = -2.294694969 + 1.875 Combine like terms: -2.294694969 + 1.875 = -0.419694969 t = -0.419694969 Simplifying t = -0.419694969Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.169694969, -0.419694969}
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