To subtract and simplify the expression [tex]\(\frac{w}{w^2 - 7w + 12} - \frac{4}{w^2 - 7w + 12}\)[/tex], let's follow a step-by-step approach:
1. Identify the Common Denominator:
The two fractions have a common denominator, which is the quadratic expression [tex]\(w^2 - 7w + 12\)[/tex].
2. Combine the Numerators Over the Common Denominator:
Since the denominators are the same, we can combine the numerators directly:
[tex]\[
\frac{w}{w^2 - 7w + 12} - \frac{4}{w^2 - 7w + 12} = \frac{w - 4}{w^2 - 7w + 12}
\][/tex]
3. Factor the Denominator:
Next, we need to simplify the fraction further by factoring the quadratic expression in the denominator. The quadratic [tex]\(w^2 - 7w + 12\)[/tex] can be factored as [tex]\((w - 3)(w - 4)\)[/tex]:
[tex]\[
w^2 - 7w + 12 = (w - 3)(w - 4)
\][/tex]
So the expression now is:
[tex]\[
\frac{w - 4}{(w - 3)(w - 4)}
\][/tex]
4. Simplify the Fraction:
Notice that [tex]\(w - 4\)[/tex] in the numerator and denominator cancels out (as long as [tex]\(w \neq 4\)[/tex], to avoid division by zero):
[tex]\[
\frac{w - 4}{(w - 3)(w - 4)} = \frac{1}{w - 3}, \quad \text{for} \quad w \neq 4
\][/tex]
So, the simplified form of the expression [tex]\(\frac{w}{w^2 - 7w + 12} - \frac{4}{w^2 - 7w + 12}\)[/tex] is:
[tex]\[
\boxed{\frac{1}{w - 3}}
\][/tex]
This is the simplest form of the given expression.
