At IDNLearn.com, find answers to your most pressing questions from experts and enthusiasts alike. Our platform is designed to provide trustworthy and thorough answers to any questions you may have.

For what values of [tex]x[/tex] is the rational expression below undefined?

[tex] \frac{x+8}{x^2 - 2x - 24} [/tex]

Check all that apply.

A. 6
B. -8
C. -4
D. 4
E. -6
F. 8


Sagot :

To determine the values of [tex]\( x \)[/tex] for which the given rational expression
[tex]\[ \frac{x+8}{x^2 - 2x - 24} \][/tex]
is undefined, we need to examine the denominator, [tex]\( x^2 - 2x - 24 \)[/tex]. The rational expression is undefined when the denominator is equal to zero, because division by zero is undefined.

Step-by-step, we solve for [tex]\( x \)[/tex] in the equation:
[tex]\[ x^2 - 2x - 24 = 0 \][/tex]

This is a quadratic equation, which we can solve by factoring. To factor [tex]\( x^2 - 2x - 24 \)[/tex], we look for two numbers that multiply to [tex]\(-24\)[/tex] and add to [tex]\(-2\)[/tex]. The numbers that fit these criteria are [tex]\( -6 \)[/tex] and [tex]\( 4 \)[/tex].

Thus, we can factor the quadratic as follows:
[tex]\[ x^2 - 2x - 24 = (x - 6)(x + 4) = 0 \][/tex]

Next, we set each factor equal to zero to solve for [tex]\( x \)[/tex]:
[tex]\[ x - 6 = 0 \quad \text{or} \quad x + 4 = 0 \][/tex]
[tex]\[ x = 6 \quad \text{or} \quad x = -4 \][/tex]

Therefore, the rational expression [tex]\(\frac{x+8}{x^2 - 2x - 24}\)[/tex] is undefined for [tex]\( x = 6 \)[/tex] and [tex]\( x = -4 \)[/tex].

So, the values of [tex]\( x \)[/tex] that make the expression undefined are:
[tex]\[ \boxed{6 \text{ and } -4} \][/tex]

From the provided options, we should select options:
A. 6
C. -4