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Simplifying 10n2 + 29n + -10 = 0 Reorder the terms: -10 + 29n + 10n2 = 0 Solving -10 + 29n + 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 + 2.9n + n2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 2.9n + 1 + n2 = 0 + 1 Reorder the terms: -1 + 1 + 2.9n + n2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 2.9n + n2 = 0 + 1 2.9n + n2 = 0 + 1 Combine like terms: 0 + 1 = 1 2.9n + n2 = 1 The n term is 2.9n. Take half its coefficient (1.45). Square it (2.1025) and add it to both sides. Add '2.1025' to each side of the equation. 2.9n + 2.1025 + n2 = 1 + 2.1025 Reorder the terms: 2.1025 + 2.9n + n2 = 1 + 2.1025 Combine like terms: 1 + 2.1025 = 3.1025 2.1025 + 2.9n + n2 = 3.1025 Factor a perfect square on the left side: (n + 1.45)(n + 1.45) = 3.1025 Calculate the square root of the right side: 1.761391495 Break this problem into two subproblems by setting (n + 1.45) equal to 1.761391495 and -1.761391495.Subproblem 1
n + 1.45 = 1.761391495 Simplifying n + 1.45 = 1.761391495 Reorder the terms: 1.45 + n = 1.761391495 Solving 1.45 + n = 1.761391495 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.45' to each side of the equation. 1.45 + -1.45 + n = 1.761391495 + -1.45 Combine like terms: 1.45 + -1.45 = 0.00 0.00 + n = 1.761391495 + -1.45 n = 1.761391495 + -1.45 Combine like terms: 1.761391495 + -1.45 = 0.311391495 n = 0.311391495 Simplifying n = 0.311391495Subproblem 2
n + 1.45 = -1.761391495 Simplifying n + 1.45 = -1.761391495 Reorder the terms: 1.45 + n = -1.761391495 Solving 1.45 + n = -1.761391495 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.45' to each side of the equation. 1.45 + -1.45 + n = -1.761391495 + -1.45 Combine like terms: 1.45 + -1.45 = 0.00 0.00 + n = -1.761391495 + -1.45 n = -1.761391495 + -1.45 Combine like terms: -1.761391495 + -1.45 = -3.211391495 n = -3.211391495 Simplifying n = -3.211391495Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.311391495, -3.211391495}
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