Question

Math

Posted 2 months ago

```
Multiply.
$\frac{x+3}{x^{2}-5 x+6} \cdot \frac{4 x-12}{x+2}$
Simplify your answer as much as possible.
```

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Answer from Sia

Posted 2 months ago

Solution by Steps

step 1

First, factor the denominators and numerators where possible. The expression is:
$\frac{x+3}{x^2 - 5x + 6} \cdot \frac{4x - 12}{x + 2}$

step 2

Factor the quadratic expression in the denominator:
$x^2 - 5x + 6 = (x - 2)(x + 3)$

step 3

Substitute the factored form into the expression:
$\frac{x+3}{(x-2)(x+3)} \cdot \frac{4x - 12}{x + 2}$

step 4

Simplify the numerator $4x - 12$:
$4x - 12 = 4(x - 3)$

step 5

Substitute the simplified numerator:
$\frac{x+3}{(x-2)(x+3)} \cdot \frac{4(x - 3)}{x + 2}$

step 6

Cancel out the common factors $(x + 3)$:
$\frac{1}{x-2} \cdot \frac{4(x - 3)}{x + 2}$

step 7

Multiply the remaining fractions:
$\frac{4(x - 3)}{(x - 2)(x + 2)}$

step 8

Simplify the final expression:
$\frac{4x - 12}{x^2 - 4}$

Answer

$\frac{4x - 12}{x^2 - 4}$

Key Concept

Simplification of Rational Expressions

Explanation

The key concept here is to factor both the numerator and the denominator where possible, cancel out common factors, and then simplify the resulting expression.

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