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Simplifying 2n(n + -7) + 3 = 9 Reorder the terms: 2n(-7 + n) + 3 = 9 (-7 * 2n + n * 2n) + 3 = 9 (-14n + 2n2) + 3 = 9 Reorder the terms: 3 + -14n + 2n2 = 9 Solving 3 + -14n + 2n2 = 9 Solving for variable 'n'. Reorder the terms: 3 + -9 + -14n + 2n2 = 9 + -9 Combine like terms: 3 + -9 = -6 -6 + -14n + 2n2 = 9 + -9 Combine like terms: 9 + -9 = 0 -6 + -14n + 2n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + -7n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + -7n + n2)' equal to zero and attempt to solve: Simplifying -3 + -7n + n2 = 0 Solving -3 + -7n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + -7n + 3 + n2 = 0 + 3 Reorder the terms: -3 + 3 + -7n + n2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + -7n + n2 = 0 + 3 -7n + n2 = 0 + 3 Combine like terms: 0 + 3 = 3 -7n + n2 = 3 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 = 3 + 12.25 Reorder the terms: 12.25 + -7n + n2 = 3 + 12.25 Combine like terms: 3 + 12.25 = 15.25 12.25 + -7n + n2 = 15.25 Factor a perfect square on the left side: (n + -3.5)(n + -3.5) = 15.25 Calculate the square root of the right side: 3.905124838 Break this problem into two subproblems by setting (n + -3.5) equal to 3.905124838 and -3.905124838.Subproblem 1
n + -3.5 = 3.905124838 Simplifying n + -3.5 = 3.905124838 Reorder the terms: -3.5 + n = 3.905124838 Solving -3.5 + n = 3.905124838 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 = 3.905124838 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + n = 3.905124838 + 3.5 n = 3.905124838 + 3.5 Combine like terms: 3.905124838 + 3.5 = 7.405124838 n = 7.405124838 Simplifying n = 7.405124838Subproblem 2
n + -3.5 = -3.905124838 Simplifying n + -3.5 = -3.905124838 Reorder the terms: -3.5 + n = -3.905124838 Solving -3.5 + n = -3.905124838 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 = -3.905124838 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + n = -3.905124838 + 3.5 n = -3.905124838 + 3.5 Combine like terms: -3.905124838 + 3.5 = -0.405124838 n = -0.405124838 Simplifying n = -0.405124838Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.405124838, -0.405124838}Solution
n = {7.405124838, -0.405124838}
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