Question Simplify the expression $\frac{x}{3 x+9}+\frac{8}{x^{2}+3 x}$ .

Studdy Solution

STEP 1

Assumptions
1. We are given the expression $\frac{x}{3x+9}+\frac{8}{x^2+3x}$ .
2. We need to simplify the expression by combining the terms.
3. We assume that $x \neq 0$ and $x \neq -3$ to avoid division by zero in the denominators.

STEP 2

First, we need to find a common denominator for the two fractions in order to combine them. The common denominator should be a multiple of both denominators $3x+9$ and $x^2+3x$ .

STEP 3

Factor out the common term in the denominators.
$3x+9 = 3(x+3)$ $x^2+3x = x(x+3)$

STEP 4

Now we can see that the common denominator is $3x(x+3)$ because it includes both factors $3$ and $x(x+3)$ .

STEP 5

Rewrite each fraction with the common denominator.
$\frac{x}{3x+9} = \frac{x}{3(x+3)} = \frac{x}{3x(x+3)} \cdot \frac{3x}{3x}$

STEP 6

Similarly, rewrite the second fraction with the common denominator.
$\frac{8}{x^2+3x} = \frac{8}{x(x+3)} = \frac{8}{3x(x+3)} \cdot \frac{3}{3}$

STEP 7

Now, multiply the numerators by the appropriate factors to get the common denominator.
$\frac{x}{3x+9} = \frac{3x^2}{3x(x+3)}$ $\frac{8}{x^2+3x} = \frac{24}{3x(x+3)}$

STEP 8

Combine the two fractions with the common denominator.
$\frac{x}{3x+9}+\frac{8}{x^2+3x} = \frac{3x^2}{3x(x+3)} + \frac{24}{3x(x+3)}$

STEP 9

Add the numerators of the fractions.
$\frac{3x^2}{3x(x+3)} + \frac{24}{3x(x+3)} = \frac{3x^2 + 24}{3x(x+3)}$

STEP 10

Look for common factors in the numerator that can be factored out.

STEP 11

Since there are no common factors to factor out in the numerator $3x^2 + 24$ , we can leave the numerator as it is.

STEP 12

Thus, the simplified expression is:
$\frac{3x^2 + 24}{3x(x+3)}$
This is the simplified form of the given expression.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms