Question

1. \\(\frac{x + 7}{x^2 + 5x - 14}\\)

Explanation:

Step1: Factor the denominator

We factor the quadratic expression \(x^{2}+5x - 14\). We need to find two numbers that multiply to \(- 14\) and add up to \(5\). The numbers are \(7\) and \(-2\) since \(7\times(-2)=-14\) and \(7+( - 2)=5\). So, \(x^{2}+5x - 14=(x + 7)(x-2)\)

Step2: Simplify the rational expression

The original expression is \(\frac{x + 7}{x^{2}+5x - 14}\), substituting the factored form of the denominator, we get \(\frac{x + 7}{(x + 7)(x-2)}\). Assuming \(x
eq - 7\) (to avoid division by zero), we can cancel out the common factor \((x + 7)\) from the numerator and the denominator. So the simplified form is \(\frac{1}{x-2}\) (where \(x
eq - 7\) and \(x
eq2\))

Answer:

\(\frac{1}{x - 2}\) (with the restrictions \(x
eq - 7\) and \(x
eq2\))