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Simplifying 5n2 + -3n = 1 Reorder the terms: -3n + 5n2 = 1 Solving -3n + 5n2 = 1 Solving for variable 'n'. Reorder the terms: -1 + -3n + 5n2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + -3n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.2 + -0.6n + n2 = 0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + -0.6n + 0.2 + n2 = 0 + 0.2 Reorder the terms: -0.2 + 0.2 + -0.6n + n2 = 0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + -0.6n + n2 = 0 + 0.2 -0.6n + n2 = 0 + 0.2 Combine like terms: 0 + 0.2 = 0.2 -0.6n + n2 = 0.2 The n term is -0.6n. Take half its coefficient (-0.3). Square it (0.09) and add it to both sides. Add '0.09' to each side of the equation. -0.6n + 0.09 + n2 = 0.2 + 0.09 Reorder the terms: 0.09 + -0.6n + n2 = 0.2 + 0.09 Combine like terms: 0.2 + 0.09 = 0.29 0.09 + -0.6n + n2 = 0.29 Factor a perfect square on the left side: (n + -0.3)(n + -0.3) = 0.29 Calculate the square root of the right side: 0.538516481 Break this problem into two subproblems by setting (n + -0.3) equal to 0.538516481 and -0.538516481.Subproblem 1
n + -0.3 = 0.538516481 Simplifying n + -0.3 = 0.538516481 Reorder the terms: -0.3 + n = 0.538516481 Solving -0.3 + n = 0.538516481 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + n = 0.538516481 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + n = 0.538516481 + 0.3 n = 0.538516481 + 0.3 Combine like terms: 0.538516481 + 0.3 = 0.838516481 n = 0.838516481 Simplifying n = 0.838516481Subproblem 2
n + -0.3 = -0.538516481 Simplifying n + -0.3 = -0.538516481 Reorder the terms: -0.3 + n = -0.538516481 Solving -0.3 + n = -0.538516481 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + n = -0.538516481 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + n = -0.538516481 + 0.3 n = -0.538516481 + 0.3 Combine like terms: -0.538516481 + 0.3 = -0.238516481 n = -0.238516481 Simplifying n = -0.238516481Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.838516481, -0.238516481}
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