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Simplifying 3n(n + -5) + -7(3n + 9) = 0 Reorder the terms: 3n(-5 + n) + -7(3n + 9) = 0 (-5 * 3n + n * 3n) + -7(3n + 9) = 0 (-15n + 3n2) + -7(3n + 9) = 0 Reorder the terms: -15n + 3n2 + -7(9 + 3n) = 0 -15n + 3n2 + (9 * -7 + 3n * -7) = 0 -15n + 3n2 + (-63 + -21n) = 0 Reorder the terms: -63 + -15n + -21n + 3n2 = 0 Combine like terms: -15n + -21n = -36n -63 + -36n + 3n2 = 0 Solving -63 + -36n + 3n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '3'. 3(-21 + -12n + n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-21 + -12n + n2)' equal to zero and attempt to solve: Simplifying -21 + -12n + n2 = 0 Solving -21 + -12n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '21' to each side of the equation. -21 + -12n + 21 + n2 = 0 + 21 Reorder the terms: -21 + 21 + -12n + n2 = 0 + 21 Combine like terms: -21 + 21 = 0 0 + -12n + n2 = 0 + 21 -12n + n2 = 0 + 21 Combine like terms: 0 + 21 = 21 -12n + n2 = 21 The n term is -12n. Take half its coefficient (-6). Square it (36) and add it to both sides. Add '36' to each side of the equation. -12n + 36 + n2 = 21 + 36 Reorder the terms: 36 + -12n + n2 = 21 + 36 Combine like terms: 21 + 36 = 57 36 + -12n + n2 = 57 Factor a perfect square on the left side: (n + -6)(n + -6) = 57 Calculate the square root of the right side: 7.549834435 Break this problem into two subproblems by setting (n + -6) equal to 7.549834435 and -7.549834435.Subproblem 1
n + -6 = 7.549834435 Simplifying n + -6 = 7.549834435 Reorder the terms: -6 + n = 7.549834435 Solving -6 + n = 7.549834435 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + n = 7.549834435 + 6 Combine like terms: -6 + 6 = 0 0 + n = 7.549834435 + 6 n = 7.549834435 + 6 Combine like terms: 7.549834435 + 6 = 13.549834435 n = 13.549834435 Simplifying n = 13.549834435Subproblem 2
n + -6 = -7.549834435 Simplifying n + -6 = -7.549834435 Reorder the terms: -6 + n = -7.549834435 Solving -6 + n = -7.549834435 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + n = -7.549834435 + 6 Combine like terms: -6 + 6 = 0 0 + n = -7.549834435 + 6 n = -7.549834435 + 6 Combine like terms: -7.549834435 + 6 = -1.549834435 n = -1.549834435 Simplifying n = -1.549834435Solution
The solution to the problem is based on the solutions from the subproblems. n = {13.549834435, -1.549834435}Solution
n = {13.549834435, -1.549834435}
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