Question

identify the nature of the quadratic equation \\(x^2 + 9x + 10 = 0\\).

• real and equal
• real and distinct

Explanation:

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<reasoning>

Identify the coefficients

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} &a = 1\\ &b = 9\\ &c = 10 \end{aligned}$$

\]

Calculate the discriminant

Using the Discriminant of Quadratic knowledge point
\[

$$\begin{aligned} \Delta &= b^2 - 4ac\\ &= 9^2 - 4(1)(10)\\ &= 81 - 40\\ &= 41 \end{aligned}$$

\]

Determine the nature of the roots

Using the Number of Real Solutions knowledge point
\[

$$\begin{aligned} &\Delta = 41 > 0 \implies \text{Two distinct real roots} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>Real and equal</mcq-option>
<mcq-correct>Real and distinct</mcq-correct>
</answer>

<post_analysis>
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"Algebra",
"Discriminant of Quadratic"
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Answer:

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<reasoning>

Identify the coefficients

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} &a = 1\\ &b = 9\\ &c = 10 \end{aligned}$$

\]

Calculate the discriminant

Using the Discriminant of Quadratic knowledge point
\[

$$\begin{aligned} \Delta &= b^2 - 4ac\\ &= 9^2 - 4(1)(10)\\ &= 81 - 40\\ &= 41 \end{aligned}$$

\]

Determine the nature of the roots

Using the Number of Real Solutions knowledge point
\[

$$\begin{aligned} &\Delta = 41 > 0 \implies \text{Two distinct real roots} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>Real and equal</mcq-option>
<mcq-correct>Real and distinct</mcq-correct>
</answer>

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