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Simplifying n2 + -1n + -21 = 72 Reorder the terms: -21 + -1n + n2 = 72 Solving -21 + -1n + n2 = 72 Solving for variable 'n'. Reorder the terms: -21 + -72 + -1n + n2 = 72 + -72 Combine like terms: -21 + -72 = -93 -93 + -1n + n2 = 72 + -72 Combine like terms: 72 + -72 = 0 -93 + -1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '93' to each side of the equation. -93 + -1n + 93 + n2 = 0 + 93 Reorder the terms: -93 + 93 + -1n + n2 = 0 + 93 Combine like terms: -93 + 93 = 0 0 + -1n + n2 = 0 + 93 -1n + n2 = 0 + 93 Combine like terms: 0 + 93 = 93 -1n + n2 = 93 The n term is -1n. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1n + 0.25 + n2 = 93 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 93 + 0.25 Combine like terms: 93 + 0.25 = 93.25 0.25 + -1n + n2 = 93.25 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 93.25 Calculate the square root of the right side: 9.656603958 Break this problem into two subproblems by setting (n + -0.5) equal to 9.656603958 and -9.656603958.Subproblem 1
n + -0.5 = 9.656603958 Simplifying n + -0.5 = 9.656603958 Reorder the terms: -0.5 + n = 9.656603958 Solving -0.5 + n = 9.656603958 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 9.656603958 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 9.656603958 + 0.5 n = 9.656603958 + 0.5 Combine like terms: 9.656603958 + 0.5 = 10.156603958 n = 10.156603958 Simplifying n = 10.156603958Subproblem 2
n + -0.5 = -9.656603958 Simplifying n + -0.5 = -9.656603958 Reorder the terms: -0.5 + n = -9.656603958 Solving -0.5 + n = -9.656603958 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -9.656603958 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -9.656603958 + 0.5 n = -9.656603958 + 0.5 Combine like terms: -9.656603958 + 0.5 = -9.156603958 n = -9.156603958 Simplifying n = -9.156603958Solution
The solution to the problem is based on the solutions from the subproblems. n = {10.156603958, -9.156603958}
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