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Simplifying -4n2 + -10n + 9 = 0 Reorder the terms: 9 + -10n + -4n2 = 0 Solving 9 + -10n + -4n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -2.25 + 2.5n + n2 = 0 Move the constant term to the right: Add '2.25' to each side of the equation. -2.25 + 2.5n + 2.25 + n2 = 0 + 2.25 Reorder the terms: -2.25 + 2.25 + 2.5n + n2 = 0 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + 2.5n + n2 = 0 + 2.25 2.5n + n2 = 0 + 2.25 Combine like terms: 0 + 2.25 = 2.25 2.5n + n2 = 2.25 The n term is 2.5n. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5n + 1.5625 + n2 = 2.25 + 1.5625 Reorder the terms: 1.5625 + 2.5n + n2 = 2.25 + 1.5625 Combine like terms: 2.25 + 1.5625 = 3.8125 1.5625 + 2.5n + n2 = 3.8125 Factor a perfect square on the left side: (n + 1.25)(n + 1.25) = 3.8125 Calculate the square root of the right side: 1.952562419 Break this problem into two subproblems by setting (n + 1.25) equal to 1.952562419 and -1.952562419.Subproblem 1
n + 1.25 = 1.952562419 Simplifying n + 1.25 = 1.952562419 Reorder the terms: 1.25 + n = 1.952562419 Solving 1.25 + n = 1.952562419 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + n = 1.952562419 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = 1.952562419 + -1.25 n = 1.952562419 + -1.25 Combine like terms: 1.952562419 + -1.25 = 0.702562419 n = 0.702562419 Simplifying n = 0.702562419Subproblem 2
n + 1.25 = -1.952562419 Simplifying n + 1.25 = -1.952562419 Reorder the terms: 1.25 + n = -1.952562419 Solving 1.25 + n = -1.952562419 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + n = -1.952562419 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = -1.952562419 + -1.25 n = -1.952562419 + -1.25 Combine like terms: -1.952562419 + -1.25 = -3.202562419 n = -3.202562419 Simplifying n = -3.202562419Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.702562419, -3.202562419}
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