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Simplifying 5 + 2n(3 + -10n) = 0 5 + (3 * 2n + -10n * 2n) = 0 5 + (6n + -20n2) = 0 Solving 5 + 6n + -20n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by -20 the coefficient of the squared term: Divide each side by '-20'. -0.25 + -0.3n + n2 = 0.0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -0.3n + 0.25 + n2 = 0.0 + 0.25 Reorder the terms: -0.25 + 0.25 + -0.3n + n2 = 0.0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -0.3n + n2 = 0.0 + 0.25 -0.3n + n2 = 0.0 + 0.25 Combine like terms: 0.0 + 0.25 = 0.25 -0.3n + n2 = 0.25 The n term is -0.3n. Take half its coefficient (-0.15). Square it (0.0225) and add it to both sides. Add '0.0225' to each side of the equation. -0.3n + 0.0225 + n2 = 0.25 + 0.0225 Reorder the terms: 0.0225 + -0.3n + n2 = 0.25 + 0.0225 Combine like terms: 0.25 + 0.0225 = 0.2725 0.0225 + -0.3n + n2 = 0.2725 Factor a perfect square on the left side: (n + -0.15)(n + -0.15) = 0.2725 Calculate the square root of the right side: 0.522015325 Break this problem into two subproblems by setting (n + -0.15) equal to 0.522015325 and -0.522015325.Subproblem 1
n + -0.15 = 0.522015325 Simplifying n + -0.15 = 0.522015325 Reorder the terms: -0.15 + n = 0.522015325 Solving -0.15 + n = 0.522015325 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.15' to each side of the equation. -0.15 + 0.15 + n = 0.522015325 + 0.15 Combine like terms: -0.15 + 0.15 = 0.00 0.00 + n = 0.522015325 + 0.15 n = 0.522015325 + 0.15 Combine like terms: 0.522015325 + 0.15 = 0.672015325 n = 0.672015325 Simplifying n = 0.672015325Subproblem 2
n + -0.15 = -0.522015325 Simplifying n + -0.15 = -0.522015325 Reorder the terms: -0.15 + n = -0.522015325 Solving -0.15 + n = -0.522015325 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.15' to each side of the equation. -0.15 + 0.15 + n = -0.522015325 + 0.15 Combine like terms: -0.15 + 0.15 = 0.00 0.00 + n = -0.522015325 + 0.15 n = -0.522015325 + 0.15 Combine like terms: -0.522015325 + 0.15 = -0.372015325 n = -0.372015325 Simplifying n = -0.372015325Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.672015325, -0.372015325}
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