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Simplifying 4n2 + -1n + -9 = -5 Reorder the terms: -9 + -1n + 4n2 = -5 Solving -9 + -1n + 4n2 = -5 Solving for variable 'n'. Reorder the terms: -9 + 5 + -1n + 4n2 = -5 + 5 Combine like terms: -9 + 5 = -4 -4 + -1n + 4n2 = -5 + 5 Combine like terms: -5 + 5 = 0 -4 + -1n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1 + -0.25n + n2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.25n + 1 + n2 = 0 + 1 Reorder the terms: -1 + 1 + -0.25n + n2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.25n + n2 = 0 + 1 -0.25n + n2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.25n + n2 = 1 The n term is -0.25n. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25n + 0.015625 + n2 = 1 + 0.015625 Reorder the terms: 0.015625 + -0.25n + n2 = 1 + 0.015625 Combine like terms: 1 + 0.015625 = 1.015625 0.015625 + -0.25n + n2 = 1.015625 Factor a perfect square on the left side: (n + -0.125)(n + -0.125) = 1.015625 Calculate the square root of the right side: 1.007782219 Break this problem into two subproblems by setting (n + -0.125) equal to 1.007782219 and -1.007782219.Subproblem 1
n + -0.125 = 1.007782219 Simplifying n + -0.125 = 1.007782219 Reorder the terms: -0.125 + n = 1.007782219 Solving -0.125 + n = 1.007782219 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + n = 1.007782219 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + n = 1.007782219 + 0.125 n = 1.007782219 + 0.125 Combine like terms: 1.007782219 + 0.125 = 1.132782219 n = 1.132782219 Simplifying n = 1.132782219Subproblem 2
n + -0.125 = -1.007782219 Simplifying n + -0.125 = -1.007782219 Reorder the terms: -0.125 + n = -1.007782219 Solving -0.125 + n = -1.007782219 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + n = -1.007782219 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + n = -1.007782219 + 0.125 n = -1.007782219 + 0.125 Combine like terms: -1.007782219 + 0.125 = -0.882782219 n = -0.882782219 Simplifying n = -0.882782219Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.132782219, -0.882782219}
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