What are the excluded values of $x$ for $\frac{x^2-3 x-28}{x^2-2 x-35}$ ?
The rational expression given is \(\frac{x^2-3x-28}{x^2-2x-35}\). For a rational expression to be defined, the denominator cannot be equal to zero. Therefore, we must identify the values of \(x\) that make the denominator \(x^2-2x-35\) equal to zero. These are the "excluded values" for the expression.
We set the denominator to zero and solve the resulting quadratic equation:
\[x^2 - 2x - 35 = 0.\]
To solve this, we look for two numbers that multiply to \(-35\) and add to \(-2\). These numbers are \(-7\) and \(5\). Thus, we can factor the quadratic as:
\[(x - 7)(x + 5) = 0.\]
Setting each factor equal to zero, we get:
\(x - 7 = 0 \implies x = 7\)
\(x + 5 = 0 \implies x = -5\)
These are the values of \(x\) that cause division by zero, making the original expression undefined. Thus, the excluded values are \(x = 7\) and \(x = -5\). Comparing this to the given options, this matches the choice \(\boxed{x=-5, x=7}\).
Excluded values are critical in fields like engineering and data science where functions model physical systems or relationships. For example, if you are modeling the efficiency of a machine as a function of temperature, and the formula involves a denominator that equals zero at a specific temperature (like \(0^\circ \text{C}\)), you must exclude that point because the machine cannot operate at that condition. Identifying these undefined points ensures that your mathematical model remains valid within its operational domain.