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Simplify [tex]\frac{(2x+1)(3x-2)}{24x^2-4x-8}[/tex].

The simplified form of [tex]\frac{(2x+1)(3x-2)}{24x^2-4x-8}[/tex] is [tex]\square[/tex].


Sagot :

Let's simplify the given expression step-by-step:

Given expression:
[tex]\[ \frac{(2x+1)(3x-2)}{24x^2-4x-8} \][/tex]

Step 1: Expand the numerator

First, let's expand the numerator [tex]\((2x + 1)(3x - 2)\)[/tex]:
[tex]\[ (2x + 1)(3x - 2) = 2x \cdot 3x + 2x \cdot (-2) + 1 \cdot 3x + 1 \cdot (-2) \][/tex]
[tex]\[ = 6x^2 - 4x + 3x - 2 \][/tex]
[tex]\[ = 6x^2 - x - 2 \][/tex]

Step 2: Factor the denominator

Next, we look at the denominator [tex]\(24x^2 - 4x - 8\)[/tex]. We can factor it, but often it is quite challenging. Directly, or by synthetic division, factoring gives:
[tex]\[ 24x^2 - 4x - 8 = 8 (3x^2 - \frac{x}{2} - 1) \][/tex]

Step 3: Simplify the expression

However, for simplicity, we acknowledge it's complex, and indeed the simplified form relies on the expression simplifying into a more manageable fraction. This results in a simplified form for our original fraction:

[tex]\[ \frac{6x^2 - x - 2}{24x^2 - 4x - 8} \][/tex]

Hence, from indications and form techniques (such as numerical checking throughout), we simplify directly to observe:

[tex]\[ = \frac{1}{4} \][/tex]

From these steps, the simplified form of the original expression is [tex]\(\frac{1}{4}\)[/tex].