Question

using the x method to factor a trinomial

what is the factored form of (6x^2 + 13x + 6)?

(x + 4)(x + 9)
(2x + 3)(3x + 2)
(2x + 6)(3x + 1)
(6x + 1)(6x + 1)

Explanation:

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<reasoning>

Identify the coefficients of the trinomial

For the trinomial \(6x^2 + 13x + 6\), compare it to the standard form \(ax^2 + bx + c\):
\[
a = 6,\quad b = 13,\quad c = 6
\]

Apply the X method parameters

Find the product \(ac\) and the sum \(b\):
\[

$$\begin{aligned} ac &= 6 \times 6 = 36\\ b &= 13 \end{aligned}$$

\]
We need two numbers that multiply to \(36\) and add to \(13\). These numbers are \(9\) and \(4\):
\[

$$\begin{aligned} 9 \times 4 &= 36\\ 9 + 4 &= 13 \end{aligned}$$

\]

Rewrite and factor by grouping

Rewrite the middle term \(13x\) as \(9x + 4x\) and factor by grouping:
\[

$$\begin{aligned} 6x^2 + 13x + 6 &= 6x^2 + 9x + 4x + 6\\ &= 3x(2x + 3) + 2(2x + 3)\\ &= (2x + 3)(3x + 2) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \((x + 4)(x + 9)\)</mcq-option>
<mcq-correct>(B) \((2x + 3)(3x + 2)\)</mcq-correct>
<mcq-option>(C) \((2x + 6)(3x + 1)\)</mcq-option>
<mcq-option>(D) \((6x + 1)(6x + 1)\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the coefficients of the trinomial

For the trinomial \(6x^2 + 13x + 6\), compare it to the standard form \(ax^2 + bx + c\):
\[
a = 6,\quad b = 13,\quad c = 6
\]

Apply the X method parameters

Find the product \(ac\) and the sum \(b\):
\[

$$\begin{aligned} ac &= 6 \times 6 = 36\\ b &= 13 \end{aligned}$$

\]
We need two numbers that multiply to \(36\) and add to \(13\). These numbers are \(9\) and \(4\):
\[

$$\begin{aligned} 9 \times 4 &= 36\\ 9 + 4 &= 13 \end{aligned}$$

\]

Rewrite and factor by grouping

Rewrite the middle term \(13x\) as \(9x + 4x\) and factor by grouping:
\[

$$\begin{aligned} 6x^2 + 13x + 6 &= 6x^2 + 9x + 4x + 6\\ &= 3x(2x + 3) + 2(2x + 3)\\ &= (2x + 3)(3x + 2) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \((x + 4)(x + 9)\)</mcq-option>
<mcq-correct>(B) \((2x + 3)(3x + 2)\)</mcq-correct>
<mcq-option>(C) \((2x + 6)(3x + 1)\)</mcq-option>
<mcq-option>(D) \((6x + 1)(6x + 1)\)</mcq-option>
</answer>

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