Question

Simplify the following rational expressions by factoring and canceling. And define the constraints (what 𝓧 cannot be) (show all work):
38x-48x2-14x+48x-8

Answer 1

To simplify the rational expression \( 38x - 48x^2 - 14x + 48x - 8 \), we will first combine like terms and then factor the result.

### Step 1: Combine Like Terms

Let's first group and combine the \( x \) terms:

\[
38x - 14x + 48x = (38 - 14 + 48)x = 72x
\]

Now, include the \( -48x^2 \) and the constant \( -8 \):

\[
-48x^2 + 72x - 8
\]

### Step 2: Factor the Expression

Now we want to factor the expression \( -48x^2 + 72x - 8 \). We can factor out the greatest common factor (GCF) of the coefficients, which is \( -8 \):

\[
-8(6x^2 - 9x + 1)
\]

Next, we will try to factor the quadratic expression \( 6x^2 - 9x + 1 \).

### Step 3: Factor the Quadratic

We can use the method of "splitting the middle term." We need to find two numbers that multiply to \( ac = 6 \times 1 = 6 \) and add to \( b = -9 \).

The numbers that satisfy this are not integers; however, we can use the quadratic formula to find the roots if necessary.

Using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 6, b = -9, c = 1 \):

\[
x = \frac{9 \pm \sqrt{(-9)^2 - 4 \cdot 6 \cdot 1}}{2 \cdot 6}
\]
\[
x = \frac{9 \pm \sqrt{81 - 24}}{12}
\]
\[
x = \frac{9 \pm \sqrt{57}}{12}
\]

Since \( \sqrt{57} \) cannot be simplified nicely, we can write the expression as:

\[
6x^2 - 9x + 1 = 0 \text{ does not factor neatly.}
\]

### Step 4: Final Factored Expression

Thus the expression cannot be factored more simply without the roots. The simplified expression is:

\[
-8(6x^2 - 9x + 1)
\]

### Step 5: Constraints

To identify the constraints for \( x \), we must ensure that the denominators are not zero. Since we do not have a denominator here, we would only be concerned about the roots of the quadratic expression:

Setting \( 6x^2 - 9x + 1 = 0 \):

\[
x = \frac{9 \pm \sqrt{57}}{12}
\]

These values indicate where the quadratic function crosses the x-axis, and we need to ensure we avoid these based on context.

### Summary

1. The simplified expression is:
\[
-8(6x^2 - 9x + 1)
\]
2. The constraints for \( x \):
\[
x \neq \frac{9 \pm \sqrt{57}}{12}
\]

This ensures we avoid points that make any expression (if there were any denominators) undefined.