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Simplifying 9n + -10n2 + 10 = 0 Reorder the terms: 10 + 9n + -10n2 = 0 Solving 10 + 9n + -10n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. -1 + -0.9n + n2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.9n + 1 + n2 = 0 + 1 Reorder the terms: -1 + 1 + -0.9n + n2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.9n + n2 = 0 + 1 -0.9n + n2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.9n + n2 = 1 The n term is -0.9n. Take half its coefficient (-0.45). Square it (0.2025) and add it to both sides. Add '0.2025' to each side of the equation. -0.9n + 0.2025 + n2 = 1 + 0.2025 Reorder the terms: 0.2025 + -0.9n + n2 = 1 + 0.2025 Combine like terms: 1 + 0.2025 = 1.2025 0.2025 + -0.9n + n2 = 1.2025 Factor a perfect square on the left side: (n + -0.45)(n + -0.45) = 1.2025 Calculate the square root of the right side: 1.09658561 Break this problem into two subproblems by setting (n + -0.45) equal to 1.09658561 and -1.09658561.Subproblem 1
n + -0.45 = 1.09658561 Simplifying n + -0.45 = 1.09658561 Reorder the terms: -0.45 + n = 1.09658561 Solving -0.45 + n = 1.09658561 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.45' to each side of the equation. -0.45 + 0.45 + n = 1.09658561 + 0.45 Combine like terms: -0.45 + 0.45 = 0.00 0.00 + n = 1.09658561 + 0.45 n = 1.09658561 + 0.45 Combine like terms: 1.09658561 + 0.45 = 1.54658561 n = 1.54658561 Simplifying n = 1.54658561Subproblem 2
n + -0.45 = -1.09658561 Simplifying n + -0.45 = -1.09658561 Reorder the terms: -0.45 + n = -1.09658561 Solving -0.45 + n = -1.09658561 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.45' to each side of the equation. -0.45 + 0.45 + n = -1.09658561 + 0.45 Combine like terms: -0.45 + 0.45 = 0.00 0.00 + n = -1.09658561 + 0.45 n = -1.09658561 + 0.45 Combine like terms: -1.09658561 + 0.45 = -0.64658561 n = -0.64658561 Simplifying n = -0.64658561Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.54658561, -0.64658561}
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