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Simplifying (4n + 3)(2n + -7) = 22 Reorder the terms: (3 + 4n)(2n + -7) = 22 Reorder the terms: (3 + 4n)(-7 + 2n) = 22 Multiply (3 + 4n) * (-7 + 2n) (3(-7 + 2n) + 4n * (-7 + 2n)) = 22 ((-7 * 3 + 2n * 3) + 4n * (-7 + 2n)) = 22 ((-21 + 6n) + 4n * (-7 + 2n)) = 22 (-21 + 6n + (-7 * 4n + 2n * 4n)) = 22 (-21 + 6n + (-28n + 8n2)) = 22 Combine like terms: 6n + -28n = -22n (-21 + -22n + 8n2) = 22 Solving -21 + -22n + 8n2 = 22 Solving for variable 'n'. Reorder the terms: -21 + -22 + -22n + 8n2 = 22 + -22 Combine like terms: -21 + -22 = -43 -43 + -22n + 8n2 = 22 + -22 Combine like terms: 22 + -22 = 0 -43 + -22n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -5.375 + -2.75n + n2 = 0 Move the constant term to the right: Add '5.375' to each side of the equation. -5.375 + -2.75n + 5.375 + n2 = 0 + 5.375 Reorder the terms: -5.375 + 5.375 + -2.75n + n2 = 0 + 5.375 Combine like terms: -5.375 + 5.375 = 0.000 0.000 + -2.75n + n2 = 0 + 5.375 -2.75n + n2 = 0 + 5.375 Combine like terms: 0 + 5.375 = 5.375 -2.75n + n2 = 5.375 The n term is -2.75n. Take half its coefficient (-1.375). Square it (1.890625) and add it to both sides. Add '1.890625' to each side of the equation. -2.75n + 1.890625 + n2 = 5.375 + 1.890625 Reorder the terms: 1.890625 + -2.75n + n2 = 5.375 + 1.890625 Combine like terms: 5.375 + 1.890625 = 7.265625 1.890625 + -2.75n + n2 = 7.265625 Factor a perfect square on the left side: (n + -1.375)(n + -1.375) = 7.265625 Calculate the square root of the right side: 2.695482332 Break this problem into two subproblems by setting (n + -1.375) equal to 2.695482332 and -2.695482332.Subproblem 1
n + -1.375 = 2.695482332 Simplifying n + -1.375 = 2.695482332 Reorder the terms: -1.375 + n = 2.695482332 Solving -1.375 + n = 2.695482332 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + n = 2.695482332 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + n = 2.695482332 + 1.375 n = 2.695482332 + 1.375 Combine like terms: 2.695482332 + 1.375 = 4.070482332 n = 4.070482332 Simplifying n = 4.070482332Subproblem 2
n + -1.375 = -2.695482332 Simplifying n + -1.375 = -2.695482332 Reorder the terms: -1.375 + n = -2.695482332 Solving -1.375 + n = -2.695482332 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + n = -2.695482332 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + n = -2.695482332 + 1.375 n = -2.695482332 + 1.375 Combine like terms: -2.695482332 + 1.375 = -1.320482332 n = -1.320482332 Simplifying n = -1.320482332Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.070482332, -1.320482332}
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