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
- The simplified expression is: \[ -8(6x^2 - 9x + 1) \]
- 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.