# Find the excluded value for the following. \frac{12a}{a^2 - 3a - 10}

## Question:

Find the excluded value for the following.

{eq}\displaystyle \frac{12a}{a^2 - 3a - 10} {/eq}

## Rational Expression:

A rational expression is a fraction where the numerator and the denominator are the expressions of a variable. A rational expression is undefined when its denominator is zero.

The given function is:

$$\frac{12a}{a^2 - 3a - 10}$$

We know that a fraction is not defined when its denominator is zero.

So for excluding values, we set the denominator to zero and solve.

$$a^2 - 3a - 10=0 \\$$

The two numbers whose product is -10 and whose sum is -3 are -5 and 2.

So we split the middle term {eq}-3a {/eq} to be {eq}-5a+2a {/eq}.

Then we get:

$$a^2-5a+2a-10=0 \\ \text{Grouping the terms,} \\ (a^2-5a)+(2a-10)=0 \\ \text{Taking 'a' and 2 as common factors from the first and the second groups,} \\ a(a-5)+2(a-5)=0 \\ \text{Taking } (a - 5) \text{ as common factor,} \\ (a-5)(a+2)=0 \\ a-5=0; \,\, a+2=0 \\ a=5; \,\, a=-2$$

Therefore, the excluded values are: {eq}\boxed{\mathbf{5 \text{ and } -2}} {/eq}. 