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Posted 2 months ago

\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: x+3x25x+64x12x+2 \frac{x+3}{x^2 - 5x + 6} \cdot \frac{4x - 12}{x + 2}
step 2
Factor the quadratic expression in the denominator: x25x+6=(x2)(x+3) x^2 - 5x + 6 = (x - 2)(x + 3)
step 3
Substitute the factored form into the expression: x+3(x2)(x+3)4x12x+2 \frac{x+3}{(x-2)(x+3)} \cdot \frac{4x - 12}{x + 2}
step 4
Simplify the numerator 4x124x - 12: 4x12=4(x3) 4x - 12 = 4(x - 3)
step 5
Substitute the simplified numerator: x+3(x2)(x+3)4(x3)x+2 \frac{x+3}{(x-2)(x+3)} \cdot \frac{4(x - 3)}{x + 2}
step 6
Cancel out the common factors (x+3)(x + 3): 1x24(x3)x+2 \frac{1}{x-2} \cdot \frac{4(x - 3)}{x + 2}
step 7
Multiply the remaining fractions: 4(x3)(x2)(x+2) \frac{4(x - 3)}{(x - 2)(x + 2)}
step 8
Simplify the final expression: 4x12x24 \frac{4x - 12}{x^2 - 4}
4x12x24\frac{4x - 12}{x^2 - 4}
Key Concept
Simplification of Rational Expressions
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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