Questions: Simplify the expression (x^2+3x-4)/(3x^2-48) (x^3-49x)/(x^2+6x-7)

Solution Steps

To simplify the given expressions, we need to factorize the numerators and denominators and then cancel out any common factors.

1. For the first expression \(\frac{x^{2}+3x-4}{3x^{2}-48}\):

   • Factorize the numerator \(x^{2}+3x-4\).
   • Factorize the denominator \(3x^{2}-48\).
   • Simplify by canceling out common factors.

2. For the second expression \(\frac{x^{3}-49x}{x^{2}+6x-7}\):

   • Factorize the numerator \(x^{3}-49x\).
   • Factorize the denominator \(x^{2}+6x-7\).
   • Simplify by canceling out common factors.

We start with the expression

\[ \frac{x^{2}+3x-4}{3x^{2}-48}. \]

First, we factor the numerator:

\[ x^{2}+3x-4 = (x - 1)(x + 4). \]

Next, we factor the denominator:

\[ 3x^{2}-48 = 3(x - 4)(x + 4). \]

Now, substituting the factored forms back into the expression gives us:

\[ \frac{(x - 1)(x + 4)}{3(x - 4)(x + 4)}. \]

We can cancel the common factor \((x + 4)\):

\[ \frac{x - 1}{3(x - 4)}. \]

Next, we simplify the expression

\[ \frac{x^{3}-49x}{x^{2}+6x-7}. \]

We start by factoring the numerator:

\[ x^{3}-49x = x(x - 7)(x + 7). \]

Now, we factor the denominator:

\[ x^{2}+6x-7 = (x - 1)(x + 7). \]

Substituting the factored forms back into the expression gives us:

\[ \frac{x(x - 7)(x + 7)}{(x - 1)(x + 7)}. \]

We can cancel the common factor \((x + 7)\):

\[ \frac{x(x - 7)}{x - 1}. \]

Final Answer