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Simplifying n2 + -13n + 22 = 7 Reorder the terms: 22 + -13n + n2 = 7 Solving 22 + -13n + n2 = 7 Solving for variable 'n'. Reorder the terms: 22 + -7 + -13n + n2 = 7 + -7 Combine like terms: 22 + -7 = 15 15 + -13n + n2 = 7 + -7 Combine like terms: 7 + -7 = 0 15 + -13n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-15' to each side of the equation. 15 + -13n + -15 + n2 = 0 + -15 Reorder the terms: 15 + -15 + -13n + n2 = 0 + -15 Combine like terms: 15 + -15 = 0 0 + -13n + n2 = 0 + -15 -13n + n2 = 0 + -15 Combine like terms: 0 + -15 = -15 -13n + n2 = -15 The n term is -13n. Take half its coefficient (-6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. -13n + 42.25 + n2 = -15 + 42.25 Reorder the terms: 42.25 + -13n + n2 = -15 + 42.25 Combine like terms: -15 + 42.25 = 27.25 42.25 + -13n + n2 = 27.25 Factor a perfect square on the left side: (n + -6.5)(n + -6.5) = 27.25 Calculate the square root of the right side: 5.220153254 Break this problem into two subproblems by setting (n + -6.5) equal to 5.220153254 and -5.220153254.Subproblem 1
n + -6.5 = 5.220153254 Simplifying n + -6.5 = 5.220153254 Reorder the terms: -6.5 + n = 5.220153254 Solving -6.5 + n = 5.220153254 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + n = 5.220153254 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + n = 5.220153254 + 6.5 n = 5.220153254 + 6.5 Combine like terms: 5.220153254 + 6.5 = 11.720153254 n = 11.720153254 Simplifying n = 11.720153254Subproblem 2
n + -6.5 = -5.220153254 Simplifying n + -6.5 = -5.220153254 Reorder the terms: -6.5 + n = -5.220153254 Solving -6.5 + n = -5.220153254 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + n = -5.220153254 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + n = -5.220153254 + 6.5 n = -5.220153254 + 6.5 Combine like terms: -5.220153254 + 6.5 = 1.279846746 n = 1.279846746 Simplifying n = 1.279846746Solution
The solution to the problem is based on the solutions from the subproblems. n = {11.720153254, 1.279846746}
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