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Simplifying n(7n + 12) = 20 Reorder the terms: n(12 + 7n) = 20 (12 * n + 7n * n) = 20 (12n + 7n2) = 20 Solving 12n + 7n2 = 20 Solving for variable 'n'. Reorder the terms: -20 + 12n + 7n2 = 20 + -20 Combine like terms: 20 + -20 = 0 -20 + 12n + 7n2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -2.857142857 + 1.714285714n + n2 = 0 Move the constant term to the right: Add '2.857142857' to each side of the equation. -2.857142857 + 1.714285714n + 2.857142857 + n2 = 0 + 2.857142857 Reorder the terms: -2.857142857 + 2.857142857 + 1.714285714n + n2 = 0 + 2.857142857 Combine like terms: -2.857142857 + 2.857142857 = 0.000000000 0.000000000 + 1.714285714n + n2 = 0 + 2.857142857 1.714285714n + n2 = 0 + 2.857142857 Combine like terms: 0 + 2.857142857 = 2.857142857 1.714285714n + n2 = 2.857142857 The n term is 1.714285714n. Take half its coefficient (0.857142857). Square it (0.7346938773) and add it to both sides. Add '0.7346938773' to each side of the equation. 1.714285714n + 0.7346938773 + n2 = 2.857142857 + 0.7346938773 Reorder the terms: 0.7346938773 + 1.714285714n + n2 = 2.857142857 + 0.7346938773 Combine like terms: 2.857142857 + 0.7346938773 = 3.5918367343 0.7346938773 + 1.714285714n + n2 = 3.5918367343 Factor a perfect square on the left side: (n + 0.857142857)(n + 0.857142857) = 3.5918367343 Calculate the square root of the right side: 1.895214166 Break this problem into two subproblems by setting (n + 0.857142857) equal to 1.895214166 and -1.895214166.Subproblem 1
n + 0.857142857 = 1.895214166 Simplifying n + 0.857142857 = 1.895214166 Reorder the terms: 0.857142857 + n = 1.895214166 Solving 0.857142857 + n = 1.895214166 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.857142857' to each side of the equation. 0.857142857 + -0.857142857 + n = 1.895214166 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + n = 1.895214166 + -0.857142857 n = 1.895214166 + -0.857142857 Combine like terms: 1.895214166 + -0.857142857 = 1.038071309 n = 1.038071309 Simplifying n = 1.038071309Subproblem 2
n + 0.857142857 = -1.895214166 Simplifying n + 0.857142857 = -1.895214166 Reorder the terms: 0.857142857 + n = -1.895214166 Solving 0.857142857 + n = -1.895214166 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.857142857' to each side of the equation. 0.857142857 + -0.857142857 + n = -1.895214166 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + n = -1.895214166 + -0.857142857 n = -1.895214166 + -0.857142857 Combine like terms: -1.895214166 + -0.857142857 = -2.752357023 n = -2.752357023 Simplifying n = -2.752357023Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.038071309, -2.752357023}
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