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Simplifying 8n(3n + 7) = 3 Reorder the terms: 8n(7 + 3n) = 3 (7 * 8n + 3n * 8n) = 3 (56n + 24n2) = 3 Solving 56n + 24n2 = 3 Solving for variable 'n'. Reorder the terms: -3 + 56n + 24n2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + 56n + 24n2 = 0 Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -0.125 + 2.333333333n + n2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + 2.333333333n + 0.125 + n2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + 2.333333333n + n2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + 2.333333333n + n2 = 0 + 0.125 2.333333333n + n2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 2.333333333n + n2 = 0.125 The n term is 2.333333333n. Take half its coefficient (1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. 2.333333333n + 1.361111112 + n2 = 0.125 + 1.361111112 Reorder the terms: 1.361111112 + 2.333333333n + n2 = 0.125 + 1.361111112 Combine like terms: 0.125 + 1.361111112 = 1.486111112 1.361111112 + 2.333333333n + n2 = 1.486111112 Factor a perfect square on the left side: (n + 1.166666667)(n + 1.166666667) = 1.486111112 Calculate the square root of the right side: 1.21906157 Break this problem into two subproblems by setting (n + 1.166666667) equal to 1.21906157 and -1.21906157.Subproblem 1
n + 1.166666667 = 1.21906157 Simplifying n + 1.166666667 = 1.21906157 Reorder the terms: 1.166666667 + n = 1.21906157 Solving 1.166666667 + n = 1.21906157 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + n = 1.21906157 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + n = 1.21906157 + -1.166666667 n = 1.21906157 + -1.166666667 Combine like terms: 1.21906157 + -1.166666667 = 0.052394903 n = 0.052394903 Simplifying n = 0.052394903Subproblem 2
n + 1.166666667 = -1.21906157 Simplifying n + 1.166666667 = -1.21906157 Reorder the terms: 1.166666667 + n = -1.21906157 Solving 1.166666667 + n = -1.21906157 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + n = -1.21906157 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + n = -1.21906157 + -1.166666667 n = -1.21906157 + -1.166666667 Combine like terms: -1.21906157 + -1.166666667 = -2.385728237 n = -2.385728237 Simplifying n = -2.385728237Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.052394903, -2.385728237}
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