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Simplifying n2 + n + -78 = 0 Reorder the terms: -78 + n + n2 = 0 Solving -78 + n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '78' to each side of the equation. -78 + n + 78 + n2 = 0 + 78 Reorder the terms: -78 + 78 + n + n2 = 0 + 78 Combine like terms: -78 + 78 = 0 0 + n + n2 = 0 + 78 n + n2 = 0 + 78 Combine like terms: 0 + 78 = 78 n + n2 = 78 The n term is n. 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. n + 0.25 + n2 = 78 + 0.25 Reorder the terms: 0.25 + n + n2 = 78 + 0.25 Combine like terms: 78 + 0.25 = 78.25 0.25 + n + n2 = 78.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 78.25 Calculate the square root of the right side: 8.845903006 Break this problem into two subproblems by setting (n + 0.5) equal to 8.845903006 and -8.845903006.Subproblem 1
n + 0.5 = 8.845903006 Simplifying n + 0.5 = 8.845903006 Reorder the terms: 0.5 + n = 8.845903006 Solving 0.5 + n = 8.845903006 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 = 8.845903006 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 8.845903006 + -0.5 n = 8.845903006 + -0.5 Combine like terms: 8.845903006 + -0.5 = 8.345903006 n = 8.345903006 Simplifying n = 8.345903006Subproblem 2
n + 0.5 = -8.845903006 Simplifying n + 0.5 = -8.845903006 Reorder the terms: 0.5 + n = -8.845903006 Solving 0.5 + n = -8.845903006 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 = -8.845903006 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -8.845903006 + -0.5 n = -8.845903006 + -0.5 Combine like terms: -8.845903006 + -0.5 = -9.345903006 n = -9.345903006 Simplifying n = -9.345903006Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.345903006, -9.345903006}
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