The formula for the sum is \( S = \frac{n(n+1)}{2} \).
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- Calculate the discriminant: \( b^2 - 4ac = (-12)^2 - 4 \times 3 \times 9 = 144 - 108 = 36 \).
- So, \( x = \frac{12 \pm \sqrt{36}}{6} = \frac{12 \pm 6}{6} \).
- This gives \( x = \frac{18}{6} = 3 \) and \( x = \frac{6}{6} = 1 \).
- Set \( \frac{n(n+1)}{2} = 210 \). Multiply through by 2: \( n(n+1) = 420 \).
- Solve the quadratic equation: \( n^2 + n - 420 = 0 \).
- Using the quadratic formula: \( n = \frac{-1 \pm \sqrt{1 + 4 \times 420}}{2} = \frac{-1 \pm \sqrt{1681}}{2} \).
