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The solutions are \(x = \frac{4 + 8}{4} = 3\) and \(x = \frac{4 - 8}{4} = -1\).
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#### \(x = 3 -1\)
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2.
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The sum of the first \(n\) terms of an arithmetic sequence is given by \(S_n = \frac{n}{2}(2a + (n-1)d)\). If the first term \(a = 5\) the common difference \(d = 3\) and \(S_n = 155\) find \(n\).
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We substitute \(a = 5\) \(d = 3\) and \(S_n = 155\) into the formula:
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\(155 = \frac{n}{2}(2 \times 5 + (n-1) \times 3)\).
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Simplify inside the parentheses: \(155 = \frac{n}{2}(10 + 3n - 3) = \frac{n}{2}(3n + 7)\).
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Multiply both sides by 2: \(310 = n(3n + 7)\).
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Expand and rearrange: \(3n^2 + 7n - 310 = 0\).
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Use the quadratic formula: \(n = \frac{-7 \pm \sqrt{7^2 - 4 \times 3 \times (-310)}}{2 \times 3}\).
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Calculate the discriminant: \(\Delta = 49 + 3720 = 3769\).
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\(\sqrt{3769} = 61\) so \(n = \frac{-7 \pm 61}{6}\).
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