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Simplifying 3n(5 + -3n) + 8 + -3 = 0 (5 * 3n + -3n * 3n) + 8 + -3 = 0 (15n + -9n2) + 8 + -3 = 0 Reorder the terms: 8 + -3 + 15n + -9n2 = 0 Combine like terms: 8 + -3 = 5 5 + 15n + -9n2 = 0 Solving 5 + 15n + -9n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -0.5555555556 + -1.666666667n + n2 = 0 Move the constant term to the right: Add '0.5555555556' to each side of the equation. -0.5555555556 + -1.666666667n + 0.5555555556 + n2 = 0 + 0.5555555556 Reorder the terms: -0.5555555556 + 0.5555555556 + -1.666666667n + n2 = 0 + 0.5555555556 Combine like terms: -0.5555555556 + 0.5555555556 = 0.0000000000 0.0000000000 + -1.666666667n + n2 = 0 + 0.5555555556 -1.666666667n + n2 = 0 + 0.5555555556 Combine like terms: 0 + 0.5555555556 = 0.5555555556 -1.666666667n + n2 = 0.5555555556 The n term is -1.666666667n. Take half its coefficient (-0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. -1.666666667n + 0.6944444447 + n2 = 0.5555555556 + 0.6944444447 Reorder the terms: 0.6944444447 + -1.666666667n + n2 = 0.5555555556 + 0.6944444447 Combine like terms: 0.5555555556 + 0.6944444447 = 1.2500000003 0.6944444447 + -1.666666667n + n2 = 1.2500000003 Factor a perfect square on the left side: (n + -0.8333333335)(n + -0.8333333335) = 1.2500000003 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (n + -0.8333333335) equal to 1.118033989 and -1.118033989.Subproblem 1
n + -0.8333333335 = 1.118033989 Simplifying n + -0.8333333335 = 1.118033989 Reorder the terms: -0.8333333335 + n = 1.118033989 Solving -0.8333333335 + n = 1.118033989 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + n = 1.118033989 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + n = 1.118033989 + 0.8333333335 n = 1.118033989 + 0.8333333335 Combine like terms: 1.118033989 + 0.8333333335 = 1.9513673225 n = 1.9513673225 Simplifying n = 1.9513673225Subproblem 2
n + -0.8333333335 = -1.118033989 Simplifying n + -0.8333333335 = -1.118033989 Reorder the terms: -0.8333333335 + n = -1.118033989 Solving -0.8333333335 + n = -1.118033989 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + n = -1.118033989 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + n = -1.118033989 + 0.8333333335 n = -1.118033989 + 0.8333333335 Combine like terms: -1.118033989 + 0.8333333335 = -0.2847006555 n = -0.2847006555 Simplifying n = -0.2847006555Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.9513673225, -0.2847006555}
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