Question

which expressions are perfect square trinomials? check all that apply.

\\(x^2 + 16x + 8\\)
\\(x^2 + 14x + 49\\)
\\(x^2 - 5x + 25\\)
\\(x^2 - 24x + 144\\)
\\(x^2 + 9x - 81\\)

Explanation:

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<reasoning>

Define perfect square trinomial criteria

\[
(x \pm a)^2 = x^2 \pm 2ax + a^2
\]

Test each given expression

\[

$$\begin{aligned} &\text{For } x^2 + 16x + 8: \quad a^2 = 8 \implies a = \sqrt{8}, \quad 2a = 4\sqrt{2} e 16 \quad (\text{Incorrect})\\ &\text{For } x^2 + 14x + 49: \quad a^2 = 49 \implies a = 7, \quad 2a = 14 \quad (\text{Correct})\\ &\text{For } x^2 - 5x + 25: \quad a^2 = 25 \implies a = 5, \quad 2a = 10 e 5 \quad (\text{Incorrect})\\ &\text{For } x^2 - 24x + 144: \quad a^2 = 144 \implies a = 12, \quad 2a = 24 \quad (\text{Correct})\\ &\text{For } x^2 + 9x - 81: \quad a^2 = -81 \quad (\text{Incorrect, constant must be positive}) \end{aligned}$$

\]

Identify correct options

\[

$$\begin{aligned} &x^2 + 14x + 49 = (x + 7)^2\\ &x^2 - 24x + 144 = (x - 12)^2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>\(x^2 + 16x + 8\)</mcq-option>
<mcq-correct>\(x^2 + 14x + 49\)</mcq-correct>
<mcq-option>\(x^2 - 5x + 25\)</mcq-option>
<mcq-correct>\(x^2 - 24x + 144\)</mcq-correct>
<mcq-option>\(x^2 + 9x - 81\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Define perfect square trinomial criteria

\[
(x \pm a)^2 = x^2 \pm 2ax + a^2
\]

Test each given expression

\[

$$\begin{aligned} &\text{For } x^2 + 16x + 8: \quad a^2 = 8 \implies a = \sqrt{8}, \quad 2a = 4\sqrt{2} e 16 \quad (\text{Incorrect})\\ &\text{For } x^2 + 14x + 49: \quad a^2 = 49 \implies a = 7, \quad 2a = 14 \quad (\text{Correct})\\ &\text{For } x^2 - 5x + 25: \quad a^2 = 25 \implies a = 5, \quad 2a = 10 e 5 \quad (\text{Incorrect})\\ &\text{For } x^2 - 24x + 144: \quad a^2 = 144 \implies a = 12, \quad 2a = 24 \quad (\text{Correct})\\ &\text{For } x^2 + 9x - 81: \quad a^2 = -81 \quad (\text{Incorrect, constant must be positive}) \end{aligned}$$

\]

Identify correct options

\[

$$\begin{aligned} &x^2 + 14x + 49 = (x + 7)^2\\ &x^2 - 24x + 144 = (x - 12)^2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>\(x^2 + 16x + 8\)</mcq-option>
<mcq-correct>\(x^2 + 14x + 49\)</mcq-correct>
<mcq-option>\(x^2 - 5x + 25\)</mcq-option>
<mcq-correct>\(x^2 - 24x + 144\)</mcq-correct>
<mcq-option>\(x^2 + 9x - 81\)</mcq-option>
</answer>

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