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Simplifying 4n + -7n(n + -9) = 42 Reorder the terms: 4n + -7n(-9 + n) = 42 4n + (-9 * -7n + n * -7n) = 42 4n + (63n + -7n2) = 42 Combine like terms: 4n + 63n = 67n 67n + -7n2 = 42 Solving 67n + -7n2 = 42 Solving for variable 'n'. Reorder the terms: -42 + 67n + -7n2 = 42 + -42 Combine like terms: 42 + -42 = 0 -42 + 67n + -7n2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. 6 + -9.571428571n + n2 = 0 Move the constant term to the right: Add '-6' to each side of the equation. 6 + -9.571428571n + -6 + n2 = 0 + -6 Reorder the terms: 6 + -6 + -9.571428571n + n2 = 0 + -6 Combine like terms: 6 + -6 = 0 0 + -9.571428571n + n2 = 0 + -6 -9.571428571n + n2 = 0 + -6 Combine like terms: 0 + -6 = -6 -9.571428571n + n2 = -6 The n term is -9.571428571n. Take half its coefficient (-4.785714286). Square it (22.90306123) and add it to both sides. Add '22.90306123' to each side of the equation. -9.571428571n + 22.90306123 + n2 = -6 + 22.90306123 Reorder the terms: 22.90306123 + -9.571428571n + n2 = -6 + 22.90306123 Combine like terms: -6 + 22.90306123 = 16.90306123 22.90306123 + -9.571428571n + n2 = 16.90306123 Factor a perfect square on the left side: (n + -4.785714286)(n + -4.785714286) = 16.90306123 Calculate the square root of the right side: 4.111333267 Break this problem into two subproblems by setting (n + -4.785714286) equal to 4.111333267 and -4.111333267.Subproblem 1
n + -4.785714286 = 4.111333267 Simplifying n + -4.785714286 = 4.111333267 Reorder the terms: -4.785714286 + n = 4.111333267 Solving -4.785714286 + n = 4.111333267 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.785714286' to each side of the equation. -4.785714286 + 4.785714286 + n = 4.111333267 + 4.785714286 Combine like terms: -4.785714286 + 4.785714286 = 0.000000000 0.000000000 + n = 4.111333267 + 4.785714286 n = 4.111333267 + 4.785714286 Combine like terms: 4.111333267 + 4.785714286 = 8.897047553 n = 8.897047553 Simplifying n = 8.897047553Subproblem 2
n + -4.785714286 = -4.111333267 Simplifying n + -4.785714286 = -4.111333267 Reorder the terms: -4.785714286 + n = -4.111333267 Solving -4.785714286 + n = -4.111333267 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.785714286' to each side of the equation. -4.785714286 + 4.785714286 + n = -4.111333267 + 4.785714286 Combine like terms: -4.785714286 + 4.785714286 = 0.000000000 0.000000000 + n = -4.111333267 + 4.785714286 n = -4.111333267 + 4.785714286 Combine like terms: -4.111333267 + 4.785714286 = 0.674381019 n = 0.674381019 Simplifying n = 0.674381019Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.897047553, 0.674381019}
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