Question

use the discriminant to determine the number and nature of the solutions to the quadratic equation. do not solve the equation.

(2x^2 + \sqrt{13}x + 3 = 0)

the discriminant is equal to .
(type an integer or a simplified fraction.)

the equation has

• two real solutions
• two nonreal solutions
• exactly one real solution

Explanation:

<pre_analysis>
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"concepts_used": [
"Quadratic Equations"
],
"new_concepts": [
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"Discriminant Analysis"
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<reasoning>

Identify coefficients of the quadratic equation

\[

$$\begin{aligned} &2x^2 + \sqrt{13}x + 3 = 0\\ &a = 2,\quad b = \sqrt{13},\quad c = 3 \end{aligned}$$

\]

Calculate the discriminant

\[

$$\begin{aligned} D &= b^2 - 4ac\\ D &= (\sqrt{13})^2 - 4(2)(3)\\ D &= 13 - 24\\ D &= -11 \end{aligned}$$

\]

Determine the number and nature of solutions

\[

$$\begin{aligned} &D = -11 < 0\\ &\text{Since } D < 0, \text{ the equation has two nonreal solutions.} \end{aligned}$$

\]
</reasoning>

<answer>
Use the discriminant to determine the number and nature of the solutions to the quadratic equation. Do not solve the equation.
\(2x^2 + \sqrt{13}x + 3 = 0\)

The discriminant is equal to <blank>-11</blank>.
(Type an integer or a simplified fraction.)

The equation has <blank>two nonreal solutions</blank>.
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Discriminant Analysis"
]
}
</post_analysis>

Answer:

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"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
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"Quadratic Equations"
],
"new_concepts": [
"Discriminant Analysis"
],
"current_concepts": [
"Quadratic Equations",
"Discriminant Analysis"
]
}
</pre_analysis>

<reasoning>

Identify coefficients of the quadratic equation

\[

$$\begin{aligned} &2x^2 + \sqrt{13}x + 3 = 0\\ &a = 2,\quad b = \sqrt{13},\quad c = 3 \end{aligned}$$

\]

Calculate the discriminant

\[

$$\begin{aligned} D &= b^2 - 4ac\\ D &= (\sqrt{13})^2 - 4(2)(3)\\ D &= 13 - 24\\ D &= -11 \end{aligned}$$

\]

Determine the number and nature of solutions

\[

$$\begin{aligned} &D = -11 < 0\\ &\text{Since } D < 0, \text{ the equation has two nonreal solutions.} \end{aligned}$$

\]
</reasoning>

<answer>
Use the discriminant to determine the number and nature of the solutions to the quadratic equation. Do not solve the equation.
\(2x^2 + \sqrt{13}x + 3 = 0\)

The discriminant is equal to <blank>-11</blank>.
(Type an integer or a simplified fraction.)

The equation has <blank>two nonreal solutions</blank>.
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Discriminant Analysis"
]
}
</post_analysis>