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Simplifying 5n2 = 9 + n Solving 5n2 = 9 + n Solving for variable 'n'. Reorder the terms: -9 + -1n + 5n2 = 9 + n + -9 + -1n Reorder the terms: -9 + -1n + 5n2 = 9 + -9 + n + -1n Combine like terms: 9 + -9 = 0 -9 + -1n + 5n2 = 0 + n + -1n -9 + -1n + 5n2 = n + -1n Combine like terms: n + -1n = 0 -9 + -1n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -1.8 + -0.2n + n2 = 0 Move the constant term to the right: Add '1.8' to each side of the equation. -1.8 + -0.2n + 1.8 + n2 = 0 + 1.8 Reorder the terms: -1.8 + 1.8 + -0.2n + n2 = 0 + 1.8 Combine like terms: -1.8 + 1.8 = 0.0 0.0 + -0.2n + n2 = 0 + 1.8 -0.2n + n2 = 0 + 1.8 Combine like terms: 0 + 1.8 = 1.8 -0.2n + n2 = 1.8 The n term is -0.2n. Take half its coefficient (-0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. -0.2n + 0.01 + n2 = 1.8 + 0.01 Reorder the terms: 0.01 + -0.2n + n2 = 1.8 + 0.01 Combine like terms: 1.8 + 0.01 = 1.81 0.01 + -0.2n + n2 = 1.81 Factor a perfect square on the left side: (n + -0.1)(n + -0.1) = 1.81 Calculate the square root of the right side: 1.345362405 Break this problem into two subproblems by setting (n + -0.1) equal to 1.345362405 and -1.345362405.Subproblem 1
n + -0.1 = 1.345362405 Simplifying n + -0.1 = 1.345362405 Reorder the terms: -0.1 + n = 1.345362405 Solving -0.1 + n = 1.345362405 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + n = 1.345362405 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + n = 1.345362405 + 0.1 n = 1.345362405 + 0.1 Combine like terms: 1.345362405 + 0.1 = 1.445362405 n = 1.445362405 Simplifying n = 1.445362405Subproblem 2
n + -0.1 = -1.345362405 Simplifying n + -0.1 = -1.345362405 Reorder the terms: -0.1 + n = -1.345362405 Solving -0.1 + n = -1.345362405 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + n = -1.345362405 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + n = -1.345362405 + 0.1 n = -1.345362405 + 0.1 Combine like terms: -1.345362405 + 0.1 = -1.245362405 n = -1.245362405 Simplifying n = -1.245362405Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.445362405, -1.245362405}
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