Question

simplify the expression \\(\frac{-b + \sqrt{b^2 - 4ac}}{2a}\\) given that \\(a = 1\\), \\(b = -18\\), and \\(c = -3\\).

\\(\frac{-b + \sqrt{b^2 - 4ac}}{2a} = \square\\)
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals and \\(i\\) as needed.)

Explanation:

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<reasoning>

Substitute the given values into the expression

\[

$$\begin{aligned} a &= 1, \quad b = -18, \quad c = -3 \\ \frac{-b + \sqrt{b^2 - 4ac}}{2a} &= \frac{-(-18) + \sqrt{(-18)^2 - 4(1)(-3)}}{2(1)} \end{aligned}$$

\]

Simplify the terms inside the radical and the denominator

\[

$$\begin{aligned} -(-18) &= 18 \\ (-18)^2 &= 324 \\ -4(1)(-3) &= 12 \\ b^2 - 4ac &= 324 + 12 = 336 \\ 2a &= 2 \\ \frac{18 + \sqrt{336}}{2} \end{aligned}$$

\]

Simplify the radical and divide by the denominator

\[

$$\begin{aligned} \sqrt{336} &= \sqrt{16 \times 21} = 4\sqrt{21} \\ \frac{18 + 4\sqrt{21}}{2} &= \frac{18}{2} + \frac{4\sqrt{21}}{2} = 9 + 2\sqrt{21} \end{aligned}$$

\]
</reasoning>

<answer>
Simplify the expression \(\frac{-b + \sqrt{b^2 - 4ac}}{2a}\) given that \(a = 1\), \(b = -18\), and \(c = -3\).

\(\frac{-b + \sqrt{b^2 - 4ac}}{2a} =\) <blank>\(9 + 2\sqrt{21}\)</blank>
</answer>

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Answer:

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<reasoning>

Substitute the given values into the expression

\[

$$\begin{aligned} a &= 1, \quad b = -18, \quad c = -3 \\ \frac{-b + \sqrt{b^2 - 4ac}}{2a} &= \frac{-(-18) + \sqrt{(-18)^2 - 4(1)(-3)}}{2(1)} \end{aligned}$$

\]

Simplify the terms inside the radical and the denominator

\[

$$\begin{aligned} -(-18) &= 18 \\ (-18)^2 &= 324 \\ -4(1)(-3) &= 12 \\ b^2 - 4ac &= 324 + 12 = 336 \\ 2a &= 2 \\ \frac{18 + \sqrt{336}}{2} \end{aligned}$$

\]

Simplify the radical and divide by the denominator

\[

$$\begin{aligned} \sqrt{336} &= \sqrt{16 \times 21} = 4\sqrt{21} \\ \frac{18 + 4\sqrt{21}}{2} &= \frac{18}{2} + \frac{4\sqrt{21}}{2} = 9 + 2\sqrt{21} \end{aligned}$$

\]
</reasoning>

<answer>
Simplify the expression \(\frac{-b + \sqrt{b^2 - 4ac}}{2a}\) given that \(a = 1\), \(b = -18\), and \(c = -3\).

\(\frac{-b + \sqrt{b^2 - 4ac}}{2a} =\) <blank>\(9 + 2\sqrt{21}\)</blank>
</answer>

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