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Simplifying 7n + 24 + -1n * n = 0 Multiply n * n 7n + 24 + -1n2 = 0 Reorder the terms: 24 + 7n + -1n2 = 0 Solving 24 + 7n + -1n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -24 + -7n + n2 = 0 Move the constant term to the right: Add '24' to each side of the equation. -24 + -7n + 24 + n2 = 0 + 24 Reorder the terms: -24 + 24 + -7n + n2 = 0 + 24 Combine like terms: -24 + 24 = 0 0 + -7n + n2 = 0 + 24 -7n + n2 = 0 + 24 Combine like terms: 0 + 24 = 24 -7n + n2 = 24 The n term is -7n. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7n + 12.25 + n2 = 24 + 12.25 Reorder the terms: 12.25 + -7n + n2 = 24 + 12.25 Combine like terms: 24 + 12.25 = 36.25 12.25 + -7n + n2 = 36.25 Factor a perfect square on the left side: (n + -3.5)(n + -3.5) = 36.25 Calculate the square root of the right side: 6.020797289 Break this problem into two subproblems by setting (n + -3.5) equal to 6.020797289 and -6.020797289.Subproblem 1
n + -3.5 = 6.020797289 Simplifying n + -3.5 = 6.020797289 Reorder the terms: -3.5 + n = 6.020797289 Solving -3.5 + n = 6.020797289 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + n = 6.020797289 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + n = 6.020797289 + 3.5 n = 6.020797289 + 3.5 Combine like terms: 6.020797289 + 3.5 = 9.520797289 n = 9.520797289 Simplifying n = 9.520797289Subproblem 2
n + -3.5 = -6.020797289 Simplifying n + -3.5 = -6.020797289 Reorder the terms: -3.5 + n = -6.020797289 Solving -3.5 + n = -6.020797289 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + n = -6.020797289 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + n = -6.020797289 + 3.5 n = -6.020797289 + 3.5 Combine like terms: -6.020797289 + 3.5 = -2.520797289 n = -2.520797289 Simplifying n = -2.520797289Solution
The solution to the problem is based on the solutions from the subproblems. n = {9.520797289, -2.520797289}
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