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Simplifying 10n(2n + 1) + -6 = 20(n + -1) + 12 Reorder the terms: 10n(1 + 2n) + -6 = 20(n + -1) + 12 (1 * 10n + 2n * 10n) + -6 = 20(n + -1) + 12 (10n + 20n2) + -6 = 20(n + -1) + 12 Reorder the terms: -6 + 10n + 20n2 = 20(n + -1) + 12 Reorder the terms: -6 + 10n + 20n2 = 20(-1 + n) + 12 -6 + 10n + 20n2 = (-1 * 20 + n * 20) + 12 -6 + 10n + 20n2 = (-20 + 20n) + 12 Reorder the terms: -6 + 10n + 20n2 = -20 + 12 + 20n Combine like terms: -20 + 12 = -8 -6 + 10n + 20n2 = -8 + 20n Solving -6 + 10n + 20n2 = -8 + 20n Solving for variable 'n'. Reorder the terms: -6 + 8 + 10n + -20n + 20n2 = -8 + 20n + 8 + -20n Combine like terms: -6 + 8 = 2 2 + 10n + -20n + 20n2 = -8 + 20n + 8 + -20n Combine like terms: 10n + -20n = -10n 2 + -10n + 20n2 = -8 + 20n + 8 + -20n Reorder the terms: 2 + -10n + 20n2 = -8 + 8 + 20n + -20n Combine like terms: -8 + 8 = 0 2 + -10n + 20n2 = 0 + 20n + -20n 2 + -10n + 20n2 = 20n + -20n Combine like terms: 20n + -20n = 0 2 + -10n + 20n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(1 + -5n + 10n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(1 + -5n + 10n2)' equal to zero and attempt to solve: Simplifying 1 + -5n + 10n2 = 0 Solving 1 + -5n + 10n2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. 0.1 + -0.5n + n2 = 0.0 Move the constant term to the right: Add '-0.1' to each side of the equation. 0.1 + -0.5n + -0.1 + n2 = 0.0 + -0.1 Reorder the terms: 0.1 + -0.1 + -0.5n + n2 = 0.0 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + -0.5n + n2 = 0.0 + -0.1 -0.5n + n2 = 0.0 + -0.1 Combine like terms: 0.0 + -0.1 = -0.1 -0.5n + n2 = -0.1 The n term is -0.5n. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5n + 0.0625 + n2 = -0.1 + 0.0625 Reorder the terms: 0.0625 + -0.5n + n2 = -0.1 + 0.0625 Combine like terms: -0.1 + 0.0625 = -0.0375 0.0625 + -0.5n + n2 = -0.0375 Factor a perfect square on the left side: (n + -0.25)(n + -0.25) = -0.0375 Can't calculate square root of the right side. The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined. The solution to this equation could not be determined.
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