Question

match the items.

a. \\(x^2 + 15 = 0\\)
b. 2
c. 1
d. 0
e. \\(x^2 - 169 = 0\\)

1. \\(x^2 + 6 = 0\\)
2. \\(x^2 - 16 = 0\\)
3. \\(x^2 + 5 = 5\\)
4. 0
5. 2

Explanation:

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<reasoning>

Determine the number of real solutions for each quadratic equation

\[

$$\begin{aligned} &\text{For } 1. \ x^2 + 6 = 0 \implies x^2 = -6 \quad (\text{No real solutions}) \implies 0 \text{ real solutions} \quad \text{(Matches d)} \\ &\text{For } 2. \ x^2 - 16 = 0 \implies x^2 = 16 \implies x = \pm 4 \quad (2 \text{ real solutions}) \implies 2 \text{ real solutions} \quad \text{(Matches b)} \\ &\text{For } 3. \ x^2 + 5 = 5 \implies x^2 = 0 \implies x = 0 \quad (1 \text{ real solution}) \implies 1 \text{ real solution} \quad \text{(Matches c)} \end{aligned}$$

\]

Match the remaining items

\[

$$\begin{aligned} &\text{For } 4. \ 0 \quad (\text{Number of real solutions for } x^2 + 15 = 0 \text{ since } x^2 = -15) \implies \text{Matches a} \\ &\text{For } 5. \ 2 \quad (\text{Number of real solutions for } x^2 - 169 = 0 \text{ since } x^2 = 169) \implies \text{Matches e} \end{aligned}$$

\]
</reasoning>

<answer>

No.	Problem	Answer
2	\(x^2 - 16 = 0\)	b
3	\(x^2 + 5 = 5\)	c
4	0	a
5	2	e

</answer>

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Answer:

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<reasoning>

Determine the number of real solutions for each quadratic equation

\[

$$\begin{aligned} &\text{For } 1. \ x^2 + 6 = 0 \implies x^2 = -6 \quad (\text{No real solutions}) \implies 0 \text{ real solutions} \quad \text{(Matches d)} \\ &\text{For } 2. \ x^2 - 16 = 0 \implies x^2 = 16 \implies x = \pm 4 \quad (2 \text{ real solutions}) \implies 2 \text{ real solutions} \quad \text{(Matches b)} \\ &\text{For } 3. \ x^2 + 5 = 5 \implies x^2 = 0 \implies x = 0 \quad (1 \text{ real solution}) \implies 1 \text{ real solution} \quad \text{(Matches c)} \end{aligned}$$

\]

Match the remaining items

\[

$$\begin{aligned} &\text{For } 4. \ 0 \quad (\text{Number of real solutions for } x^2 + 15 = 0 \text{ since } x^2 = -15) \implies \text{Matches a} \\ &\text{For } 5. \ 2 \quad (\text{Number of real solutions for } x^2 - 169 = 0 \text{ since } x^2 = 169) \implies \text{Matches e} \end{aligned}$$

\]
</reasoning>

<answer>

No.	Problem	Answer
2	\(x^2 - 16 = 0\)	b
3	\(x^2 + 5 = 5\)	c
4	0	a
5	2	e

</answer>

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