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Simplifying 0.0000000001234451[n * 123] + -1n[84 + n] + -456 = 2 Reorder the terms for easier multiplication: 0.0000000001234451[123n] + -1n[84 + n] + -456 = 2 Remove brackets around [123n] 0.0000000001234451 * 123n + -1n[84 + n] + -456 = 2 Multiply 0.0000000001234451 * 123 0.0000000151837473n + -1n[84 + n] + -456 = 2 0.0000000151837473n + [84 * -1n + n * -1n] + -456 = 2 0.0000000151837473n + [-84n + -1n2] + -456 = 2 Reorder the terms: -456 + 0.0000000151837473n + -84n + -1n2 = 2 Combine like terms: 0.0000000151837473n + -84n = -83.9999999848162527n -456 + -83.9999999848162527n + -1n2 = 2 Solving -456 + -83.9999999848162527n + -1n2 = 2 Solving for variable 'n'. Reorder the terms: -456 + -2 + -83.9999999848162527n + -1n2 = 2 + -2 Combine like terms: -456 + -2 = -458 -458 + -83.9999999848162527n + -1n2 = 2 + -2 Combine like terms: 2 + -2 = 0 -458 + -83.9999999848162527n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 458 + 83.99999998n + n2 = 0 Move the constant term to the right: Add '-458' to each side of the equation. 458 + 83.99999998n + -458 + n2 = 0 + -458 Reorder the terms: 458 + -458 + 83.99999998n + n2 = 0 + -458 Combine like terms: 458 + -458 = 0 0 + 83.99999998n + n2 = 0 + -458 83.99999998n + n2 = 0 + -458 Combine like terms: 0 + -458 = -458 83.99999998n + n2 = -458 The n term is 83.99999998n. Take half its coefficient (41.99999999). Square it (1763.999999) and add it to both sides. Add '1763.999999' to each side of the equation. 83.99999998n + 1763.999999 + n2 = -458 + 1763.999999 Reorder the terms: 1763.999999 + 83.99999998n + n2 = -458 + 1763.999999 Combine like terms: -458 + 1763.999999 = 1305.999999 1763.999999 + 83.99999998n + n2 = 1305.999999 Factor a perfect square on the left side: (n + 41.99999999)(n + 41.99999999) = 1305.999999 Calculate the square root of the right side: 36.138621985 Break this problem into two subproblems by setting (n + 41.99999999) equal to 36.138621985 and -36.138621985.Subproblem 1
n + 41.99999999 = 36.138621985 Simplifying n + 41.99999999 = 36.138621985 Reorder the terms: 41.99999999 + n = 36.138621985 Solving 41.99999999 + n = 36.138621985 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-41.99999999' to each side of the equation. 41.99999999 + -41.99999999 + n = 36.138621985 + -41.99999999 Combine like terms: 41.99999999 + -41.99999999 = 0.00000000 0.00000000 + n = 36.138621985 + -41.99999999 n = 36.138621985 + -41.99999999 Combine like terms: 36.138621985 + -41.99999999 = -5.861378005 n = -5.861378005 Simplifying n = -5.861378005Subproblem 2
n + 41.99999999 = -36.138621985 Simplifying n + 41.99999999 = -36.138621985 Reorder the terms: 41.99999999 + n = -36.138621985 Solving 41.99999999 + n = -36.138621985 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-41.99999999' to each side of the equation. 41.99999999 + -41.99999999 + n = -36.138621985 + -41.99999999 Combine like terms: 41.99999999 + -41.99999999 = 0.00000000 0.00000000 + n = -36.138621985 + -41.99999999 n = -36.138621985 + -41.99999999 Combine like terms: -36.138621985 + -41.99999999 = -78.138621975 n = -78.138621975 Simplifying n = -78.138621975Solution
The solution to the problem is based on the solutions from the subproblems. n = {-5.861378005, -78.138621975}
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