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Simplifying 2n2 + -10n + 9 = 0 Reorder the terms: 9 + -10n + 2n2 = 0 Solving 9 + -10n + 2n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 4.5 + -5n + n2 = 0 Move the constant term to the right: Add '-4.5' to each side of the equation. 4.5 + -5n + -4.5 + n2 = 0 + -4.5 Reorder the terms: 4.5 + -4.5 + -5n + n2 = 0 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + -5n + n2 = 0 + -4.5 -5n + n2 = 0 + -4.5 Combine like terms: 0 + -4.5 = -4.5 -5n + n2 = -4.5 The n term is -5n. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5n + 6.25 + n2 = -4.5 + 6.25 Reorder the terms: 6.25 + -5n + n2 = -4.5 + 6.25 Combine like terms: -4.5 + 6.25 = 1.75 6.25 + -5n + n2 = 1.75 Factor a perfect square on the left side: (n + -2.5)(n + -2.5) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (n + -2.5) equal to 1.322875656 and -1.322875656.Subproblem 1
n + -2.5 = 1.322875656 Simplifying n + -2.5 = 1.322875656 Reorder the terms: -2.5 + n = 1.322875656 Solving -2.5 + n = 1.322875656 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + n = 1.322875656 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = 1.322875656 + 2.5 n = 1.322875656 + 2.5 Combine like terms: 1.322875656 + 2.5 = 3.822875656 n = 3.822875656 Simplifying n = 3.822875656Subproblem 2
n + -2.5 = -1.322875656 Simplifying n + -2.5 = -1.322875656 Reorder the terms: -2.5 + n = -1.322875656 Solving -2.5 + n = -1.322875656 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + n = -1.322875656 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = -1.322875656 + 2.5 n = -1.322875656 + 2.5 Combine like terms: -1.322875656 + 2.5 = 1.177124344 n = 1.177124344 Simplifying n = 1.177124344Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.822875656, 1.177124344}
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