Questions: (x^2-7x+10)/(x^2+4x-12)

Transcript text: \[ \frac{x^{2}-7 x+10}{x^{2}+4 x-12} \]

Solution

Solution Steps

To simplify the given rational expression, we need to factor both the numerator and the denominator and then cancel out any common factors.

Step 1: Factor the Numerator

The numerator of the given rational expression is \(x^2 - 7x + 10\). We factor this quadratic expression: \[ x^2 - 7x + 10 = (x - 5)(x - 2) \]

Step 2: Factor the Denominator

The denominator of the given rational expression is \(x^2 + 4x - 12\). We factor this quadratic expression: \[ x^2 + 4x - 12 = (x - 2)(x + 6) \]

Step 3: Simplify the Rational Expression

We now have the factored form of the rational expression: \[ \frac{x^2 - 7x + 10}{x^2 + 4x - 12} = \frac{(x - 5)(x - 2)}{(x - 2)(x + 6)} \] We can cancel the common factor \((x - 2)\) from the numerator and the denominator: \[ \frac{(x - 5)(x - 2)}{(x - 2)(x + 6)} = \frac{x - 5}{x + 6} \]

Final Answer

The simplified form of the given rational expression is: \[ \boxed{\frac{x - 5}{x + 6}} \]

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