Question
Simplify. Express your answer as a single fraction in simplest form.
s
s2+24s–23
+8
s
s2+24s–23
+8
Answers
GPT-4o mini
To simplify the expression
\[
\frac{s}{s^2 + 24s - 23} + 8,
\]
we first express \( 8 \) as a fraction with the same denominator:
\[
8 = \frac{8(s^2 + 24s - 23)}{s^2 + 24s - 23}.
\]
Now, we can combine the fractions:
\[
\frac{s}{s^2 + 24s - 23} + \frac{8(s^2 + 24s - 23)}{s^2 + 24s - 23} = \frac{s + 8(s^2 + 24s - 23)}{s^2 + 24s - 23}.
\]
Next, we simplify the numerator:
\[
s + 8(s^2 + 24s - 23) = s + 8s^2 + 192s - 184 = 8s^2 + 193s - 184.
\]
Thus, the expression becomes:
\[
\frac{8s^2 + 193s - 184}{s^2 + 24s - 23}.
\]
Now, we will check if we can factor the numerator \( 8s^2 + 193s - 184 \) and the denominator \( s^2 + 24s - 23 \) to see if they have common factors.
First, we focus on the denominator:
\[
s^2 + 24s - 23.
\]
To factor it, we are looking for two numbers that multiply to \(-23\) and add to \(24\). The numbers \(25\) and \(-1\) work:
\[
s^2 + 25s - s - 23 = (s + 25)(s - 1).
\]
Now for the numerator \( 8s^2 + 193s - 184 \), we will use the quadratic formula to find its roots:
\[
s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
\]
where \( a = 8, b = 193, c = -184 \):
\[
s = \frac{-193 \pm \sqrt{193^2 - 4 \cdot 8 \cdot (-184)}}{2 \cdot 8}.
\]
Calculating the discriminant:
\[
193^2 = 37249,
\]
\[
-4 \cdot 8 \cdot (-184) = 5856,
\]
\[
193^2 + 4 \cdot 8 \cdot 184 = 37249 + 5856 = 43105.
\]
Now we apply the formula:
\[
s = \frac{-193 \pm \sqrt{43105}}{16}.
\]
There doesn't appear to be an easy integer or simple rational root for \( 8s^2 + 193s - 184 \), thus we expect it may not factor nicely.
In conclusion, as it stands, we can express the simplified result as:
\[
\frac{8s^2 + 193s - 184}{(s + 25)(s - 1)}.
\]
While not factored completely, this is your final expression:
\[
\frac{8s^2 + 193s - 184}{s^2 + 24s - 23}.
\]
We recommend checking if the numerator \(8s^2 + 193s - 184\) can be factored or simplified further, but it appears not to have common factors with the denominator after our analysis.
\[
\frac{s}{s^2 + 24s - 23} + 8,
\]
we first express \( 8 \) as a fraction with the same denominator:
\[
8 = \frac{8(s^2 + 24s - 23)}{s^2 + 24s - 23}.
\]
Now, we can combine the fractions:
\[
\frac{s}{s^2 + 24s - 23} + \frac{8(s^2 + 24s - 23)}{s^2 + 24s - 23} = \frac{s + 8(s^2 + 24s - 23)}{s^2 + 24s - 23}.
\]
Next, we simplify the numerator:
\[
s + 8(s^2 + 24s - 23) = s + 8s^2 + 192s - 184 = 8s^2 + 193s - 184.
\]
Thus, the expression becomes:
\[
\frac{8s^2 + 193s - 184}{s^2 + 24s - 23}.
\]
Now, we will check if we can factor the numerator \( 8s^2 + 193s - 184 \) and the denominator \( s^2 + 24s - 23 \) to see if they have common factors.
First, we focus on the denominator:
\[
s^2 + 24s - 23.
\]
To factor it, we are looking for two numbers that multiply to \(-23\) and add to \(24\). The numbers \(25\) and \(-1\) work:
\[
s^2 + 25s - s - 23 = (s + 25)(s - 1).
\]
Now for the numerator \( 8s^2 + 193s - 184 \), we will use the quadratic formula to find its roots:
\[
s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
\]
where \( a = 8, b = 193, c = -184 \):
\[
s = \frac{-193 \pm \sqrt{193^2 - 4 \cdot 8 \cdot (-184)}}{2 \cdot 8}.
\]
Calculating the discriminant:
\[
193^2 = 37249,
\]
\[
-4 \cdot 8 \cdot (-184) = 5856,
\]
\[
193^2 + 4 \cdot 8 \cdot 184 = 37249 + 5856 = 43105.
\]
Now we apply the formula:
\[
s = \frac{-193 \pm \sqrt{43105}}{16}.
\]
There doesn't appear to be an easy integer or simple rational root for \( 8s^2 + 193s - 184 \), thus we expect it may not factor nicely.
In conclusion, as it stands, we can express the simplified result as:
\[
\frac{8s^2 + 193s - 184}{(s + 25)(s - 1)}.
\]
While not factored completely, this is your final expression:
\[
\frac{8s^2 + 193s - 184}{s^2 + 24s - 23}.
\]
We recommend checking if the numerator \(8s^2 + 193s - 184\) can be factored or simplified further, but it appears not to have common factors with the denominator after our analysis.