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Simplifying 0.4225 = n(n + 2) Reorder the terms: 0.4225 = n(2 + n) 0.4225 = (2 * n + n * n) 0.4225 = (2n + n2) Solving 0.4225 = 2n + n2 Solving for variable 'n'. Reorder the terms: 0.4225 + -2n + -1n2 = 2n + -2n + n2 + -1n2 Combine like terms: 2n + -2n = 0 0.4225 + -2n + -1n2 = 0 + n2 + -1n2 0.4225 + -2n + -1n2 = n2 + -1n2 Combine like terms: n2 + -1n2 = 0 0.4225 + -2n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -0.4225 + 2n + n2 = 0 Move the constant term to the right: Add '0.4225' to each side of the equation. -0.4225 + 2n + 0.4225 + n2 = 0 + 0.4225 Reorder the terms: -0.4225 + 0.4225 + 2n + n2 = 0 + 0.4225 Combine like terms: -0.4225 + 0.4225 = 0.0000 0.0000 + 2n + n2 = 0 + 0.4225 2n + n2 = 0 + 0.4225 Combine like terms: 0 + 0.4225 = 0.4225 2n + n2 = 0.4225 The n term is 2n. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2n + 1 + n2 = 0.4225 + 1 Reorder the terms: 1 + 2n + n2 = 0.4225 + 1 Combine like terms: 0.4225 + 1 = 1.4225 1 + 2n + n2 = 1.4225 Factor a perfect square on the left side: (n + 1)(n + 1) = 1.4225 Calculate the square root of the right side: 1.192686044 Break this problem into two subproblems by setting (n + 1) equal to 1.192686044 and -1.192686044.Subproblem 1
n + 1 = 1.192686044 Simplifying n + 1 = 1.192686044 Reorder the terms: 1 + n = 1.192686044 Solving 1 + n = 1.192686044 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 1.192686044 + -1 Combine like terms: 1 + -1 = 0 0 + n = 1.192686044 + -1 n = 1.192686044 + -1 Combine like terms: 1.192686044 + -1 = 0.192686044 n = 0.192686044 Simplifying n = 0.192686044Subproblem 2
n + 1 = -1.192686044 Simplifying n + 1 = -1.192686044 Reorder the terms: 1 + n = -1.192686044 Solving 1 + n = -1.192686044 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -1.192686044 + -1 Combine like terms: 1 + -1 = 0 0 + n = -1.192686044 + -1 n = -1.192686044 + -1 Combine like terms: -1.192686044 + -1 = -2.192686044 n = -2.192686044 Simplifying n = -2.192686044Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.192686044, -2.192686044}
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