Question

which equation has two roots?

\\(x^2 - 12x + 20 = 0\\)

\\(x + 8 = 0\\)

\\(x^3 - 12x + 20 = 0\\)

all of the choices

Explanation:

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<reasoning>

Analyze the degree of each polynomial equation

Using the Degree of Polynomial and Fundamental Theorem of Algebra knowledge points
\[

$$\begin{aligned} &\text{Degree of } x^2 - 12x + 20 = 0 \text{ is } 2 \implies 2 \text{ complex roots}\\ &\text{Degree of } x + 8 = 0 \text{ is } 1 \implies 1 \text{ root}\\ &\text{Degree of } x^3 - 12x + 20 = 0 \text{ is } 3 \implies 3 \text{ complex roots} \end{aligned}$$

\]

Determine the number of real roots for the quadratic equation

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} &x^2 - 12x + 20 = 0\\ &(x - 10)(x - 2) = 0\\ &x = 10 \quad \text{or} \quad x = 2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(x^2 - 12x + 20 = 0\)</mcq-correct>
<mcq-option>(B) \(x + 8 = 0\)</mcq-option>
<mcq-option>(C) \(x^3 - 12x + 20 = 0\)</mcq-option>
<mcq-option>(D) All of the choices</mcq-option>
</answer>

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Answer:

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<reasoning>

Analyze the degree of each polynomial equation

Using the Degree of Polynomial and Fundamental Theorem of Algebra knowledge points
\[

$$\begin{aligned} &\text{Degree of } x^2 - 12x + 20 = 0 \text{ is } 2 \implies 2 \text{ complex roots}\\ &\text{Degree of } x + 8 = 0 \text{ is } 1 \implies 1 \text{ root}\\ &\text{Degree of } x^3 - 12x + 20 = 0 \text{ is } 3 \implies 3 \text{ complex roots} \end{aligned}$$

\]

Determine the number of real roots for the quadratic equation

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} &x^2 - 12x + 20 = 0\\ &(x - 10)(x - 2) = 0\\ &x = 10 \quad \text{or} \quad x = 2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(x^2 - 12x + 20 = 0\)</mcq-correct>
<mcq-option>(B) \(x + 8 = 0\)</mcq-option>
<mcq-option>(C) \(x^3 - 12x + 20 = 0\)</mcq-option>
<mcq-option>(D) All of the choices</mcq-option>
</answer>

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