To simplify the expression [tex]\(\frac{6x^2 + 35x - 6}{2x^2 - 72}\)[/tex], we follow several steps to factorize both the numerator and the denominator and simplify the overall fraction.
1. Factorize the Denominator:
The denominator is a quadratic expression [tex]\(2x^2 - 72\)[/tex].
- First, factor out the common factor of 2:
[tex]\[
2x^2 - 72 = 2(x^2 - 36)
\][/tex]
- Next, recognize that [tex]\(x^2 - 36\)[/tex] is a difference of squares, which can be factored as:
[tex]\[
x^2 - 36 = (x - 6)(x + 6)
\][/tex]
- Therefore, the denominator [tex]\(2x^2 - 72\)[/tex] can be written as:
[tex]\[
2(x^2 - 36) = 2(x - 6)(x + 6)
\][/tex]
2. Factorize the Numerator:
The numerator is a quadratic expression [tex]\(6x^2 + 35x - 6\)[/tex].
- This can be factored into two binomials. The correct factorization is:
[tex]\[
6x^2 + 35x - 6 = (3x - 1)(2x + 6)
\][/tex]
3. Simplify the Fraction:
Now that we have factored both the numerator and the denominator, the expression can be rewritten as:
[tex]\[
\frac{(3x - 1)(2x + 6)}{2(x - 6)(x + 6)}
\][/tex]
4. Cancel Out Common Factors:
In this fraction, there are no common factors between the numerator and the denominator that can be cancelled. However, we can simplify the numerator further. Notice that [tex]\(2x + 6\)[/tex] can be simplified by factoring out 2:
[tex]\[
2x + 6 = 2(x + 3)
\][/tex]
Then the numerator becomes:
[tex]\[
(3x - 1)(2(x + 3))
\][/tex]
Now the entire fraction is:
[tex]\[
\frac{(3x - 1) \cdot 2(x + 3)}{2(x - 6)(x + 6)}
\][/tex]
We see that we can cancel out the common factor of 2 in the numerator and the denominator:
[tex]\[
\frac{(3x - 1)(x + 3)}{(x - 6)(x + 6)}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\frac{6x^2 + 35x - 6}{2x^2 - 72} = \frac{3x - 1}{2(x - 6)}
\][/tex]
