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Simplifying (n + 4)(n + -3) = 22 Reorder the terms: (4 + n)(n + -3) = 22 Reorder the terms: (4 + n)(-3 + n) = 22 Multiply (4 + n) * (-3 + n) (4(-3 + n) + n(-3 + n)) = 22 ((-3 * 4 + n * 4) + n(-3 + n)) = 22 ((-12 + 4n) + n(-3 + n)) = 22 (-12 + 4n + (-3 * n + n * n)) = 22 (-12 + 4n + (-3n + n2)) = 22 Combine like terms: 4n + -3n = 1n (-12 + 1n + n2) = 22 Solving -12 + 1n + n2 = 22 Solving for variable 'n'. Reorder the terms: -12 + -22 + 1n + n2 = 22 + -22 Combine like terms: -12 + -22 = -34 -34 + 1n + n2 = 22 + -22 Combine like terms: 22 + -22 = 0 -34 + 1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '34' to each side of the equation. -34 + 1n + 34 + n2 = 0 + 34 Reorder the terms: -34 + 34 + 1n + n2 = 0 + 34 Combine like terms: -34 + 34 = 0 0 + 1n + n2 = 0 + 34 1n + n2 = 0 + 34 Combine like terms: 0 + 34 = 34 1n + n2 = 34 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. n + 0.25 + n2 = 34 + 0.25 Reorder the terms: 0.25 + n + n2 = 34 + 0.25 Combine like terms: 34 + 0.25 = 34.25 0.25 + n + n2 = 34.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 34.25 Calculate the square root of the right side: 5.852349955 Break this problem into two subproblems by setting (n + 0.5) equal to 5.852349955 and -5.852349955.Subproblem 1
n + 0.5 = 5.852349955 Simplifying n + 0.5 = 5.852349955 Reorder the terms: 0.5 + n = 5.852349955 Solving 0.5 + n = 5.852349955 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 = 5.852349955 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 5.852349955 + -0.5 n = 5.852349955 + -0.5 Combine like terms: 5.852349955 + -0.5 = 5.352349955 n = 5.352349955 Simplifying n = 5.352349955Subproblem 2
n + 0.5 = -5.852349955 Simplifying n + 0.5 = -5.852349955 Reorder the terms: 0.5 + n = -5.852349955 Solving 0.5 + n = -5.852349955 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 = -5.852349955 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -5.852349955 + -0.5 n = -5.852349955 + -0.5 Combine like terms: -5.852349955 + -0.5 = -6.352349955 n = -6.352349955 Simplifying n = -6.352349955Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.352349955, -6.352349955}
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