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Simplifying 10n2 + -11n + -15 = -8 Reorder the terms: -15 + -11n + 10n2 = -8 Solving -15 + -11n + 10n2 = -8 Solving for variable 'n'. Reorder the terms: -15 + 8 + -11n + 10n2 = -8 + 8 Combine like terms: -15 + 8 = -7 -7 + -11n + 10n2 = -8 + 8 Combine like terms: -8 + 8 = 0 -7 + -11n + 10n2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -0.7 + -1.1n + n2 = 0 Move the constant term to the right: Add '0.7' to each side of the equation. -0.7 + -1.1n + 0.7 + n2 = 0 + 0.7 Reorder the terms: -0.7 + 0.7 + -1.1n + n2 = 0 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + -1.1n + n2 = 0 + 0.7 -1.1n + n2 = 0 + 0.7 Combine like terms: 0 + 0.7 = 0.7 -1.1n + n2 = 0.7 The n term is -1.1n. Take half its coefficient (-0.55). Square it (0.3025) and add it to both sides. Add '0.3025' to each side of the equation. -1.1n + 0.3025 + n2 = 0.7 + 0.3025 Reorder the terms: 0.3025 + -1.1n + n2 = 0.7 + 0.3025 Combine like terms: 0.7 + 0.3025 = 1.0025 0.3025 + -1.1n + n2 = 1.0025 Factor a perfect square on the left side: (n + -0.55)(n + -0.55) = 1.0025 Calculate the square root of the right side: 1.00124922 Break this problem into two subproblems by setting (n + -0.55) equal to 1.00124922 and -1.00124922.Subproblem 1
n + -0.55 = 1.00124922 Simplifying n + -0.55 = 1.00124922 Reorder the terms: -0.55 + n = 1.00124922 Solving -0.55 + n = 1.00124922 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.55' to each side of the equation. -0.55 + 0.55 + n = 1.00124922 + 0.55 Combine like terms: -0.55 + 0.55 = 0.00 0.00 + n = 1.00124922 + 0.55 n = 1.00124922 + 0.55 Combine like terms: 1.00124922 + 0.55 = 1.55124922 n = 1.55124922 Simplifying n = 1.55124922Subproblem 2
n + -0.55 = -1.00124922 Simplifying n + -0.55 = -1.00124922 Reorder the terms: -0.55 + n = -1.00124922 Solving -0.55 + n = -1.00124922 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.55' to each side of the equation. -0.55 + 0.55 + n = -1.00124922 + 0.55 Combine like terms: -0.55 + 0.55 = 0.00 0.00 + n = -1.00124922 + 0.55 n = -1.00124922 + 0.55 Combine like terms: -1.00124922 + 0.55 = -0.45124922 n = -0.45124922 Simplifying n = -0.45124922Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.55124922, -0.45124922}
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