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Simplifying (4t + -7)(7t + 7) + 5t + -2 = 0 Reorder the terms: (-7 + 4t)(7t + 7) + 5t + -2 = 0 Reorder the terms: (-7 + 4t)(7 + 7t) + 5t + -2 = 0 Multiply (-7 + 4t) * (7 + 7t) (-7(7 + 7t) + 4t * (7 + 7t)) + 5t + -2 = 0 ((7 * -7 + 7t * -7) + 4t * (7 + 7t)) + 5t + -2 = 0 ((-49 + -49t) + 4t * (7 + 7t)) + 5t + -2 = 0 (-49 + -49t + (7 * 4t + 7t * 4t)) + 5t + -2 = 0 (-49 + -49t + (28t + 28t2)) + 5t + -2 = 0 Combine like terms: -49t + 28t = -21t (-49 + -21t + 28t2) + 5t + -2 = 0 Reorder the terms: -49 + -2 + -21t + 5t + 28t2 = 0 Combine like terms: -49 + -2 = -51 -51 + -21t + 5t + 28t2 = 0 Combine like terms: -21t + 5t = -16t -51 + -16t + 28t2 = 0 Solving -51 + -16t + 28t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 28 the coefficient of the squared term: Divide each side by '28'. -1.821428571 + -0.5714285714t + t2 = 0 Move the constant term to the right: Add '1.821428571' to each side of the equation. -1.821428571 + -0.5714285714t + 1.821428571 + t2 = 0 + 1.821428571 Reorder the terms: -1.821428571 + 1.821428571 + -0.5714285714t + t2 = 0 + 1.821428571 Combine like terms: -1.821428571 + 1.821428571 = 0.000000000 0.000000000 + -0.5714285714t + t2 = 0 + 1.821428571 -0.5714285714t + t2 = 0 + 1.821428571 Combine like terms: 0 + 1.821428571 = 1.821428571 -0.5714285714t + t2 = 1.821428571 The t term is -0.5714285714t. Take half its coefficient (-0.2857142857). Square it (0.08163265305) and add it to both sides. Add '0.08163265305' to each side of the equation. -0.5714285714t + 0.08163265305 + t2 = 1.821428571 + 0.08163265305 Reorder the terms: 0.08163265305 + -0.5714285714t + t2 = 1.821428571 + 0.08163265305 Combine like terms: 1.821428571 + 0.08163265305 = 1.90306122405 0.08163265305 + -0.5714285714t + t2 = 1.90306122405 Factor a perfect square on the left side: (t + -0.2857142857)(t + -0.2857142857) = 1.90306122405 Calculate the square root of the right side: 1.379514851 Break this problem into two subproblems by setting (t + -0.2857142857) equal to 1.379514851 and -1.379514851.Subproblem 1
t + -0.2857142857 = 1.379514851 Simplifying t + -0.2857142857 = 1.379514851 Reorder the terms: -0.2857142857 + t = 1.379514851 Solving -0.2857142857 + t = 1.379514851 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + t = 1.379514851 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + t = 1.379514851 + 0.2857142857 t = 1.379514851 + 0.2857142857 Combine like terms: 1.379514851 + 0.2857142857 = 1.6652291367 t = 1.6652291367 Simplifying t = 1.6652291367Subproblem 2
t + -0.2857142857 = -1.379514851 Simplifying t + -0.2857142857 = -1.379514851 Reorder the terms: -0.2857142857 + t = -1.379514851 Solving -0.2857142857 + t = -1.379514851 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + t = -1.379514851 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + t = -1.379514851 + 0.2857142857 t = -1.379514851 + 0.2857142857 Combine like terms: -1.379514851 + 0.2857142857 = -1.0938005653 t = -1.0938005653 Simplifying t = -1.0938005653Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.6652291367, -1.0938005653}
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