For what value(s) of x is the expression racx2 + 15x + 50x−5 not defined? Find the value(s) of x for which…

Question

For what value(s) of x is the expression racx2 + 15x + 50x−5 not defined? Find the value(s) of x for which the expression is zero.

ScanSolve AI Tutor · Accepted Answer

This question has two parts. Let's solve them one by one.

Question 8 (a)(i): For what value(s) of $x$ is the expression $\frac{x^2 + 15x + 50}{x-5}$ not defined?

An algebraic expression with a fraction is undefined when its denominator is equal to zero.
Set the denominator to zero:
$x - 5 = 0$
Solve for $x$ .
$x = 5$
The expression is not defined when $x = \boxed{\textbf{5}}$ .

Question 8 (a)(ii): Find the value(s) of $x$ for which the expression is zero.
An algebraic expression with a fraction is zero when its numerator is equal to zero, provided the denominator is not zero for those values of $x$ .
Set the numerator to zero:
$x^2 + 15x + 50 = 0$
Factorize the quadratic equation.
We need two numbers that multiply to 50 and add to 15. These numbers are 5 and 10.
$x^2 + 5x + 10x + 50 = 0$
Factor by grouping.
$x(x+5) + 10(x+5) = 0$ $(x+5)(x+10) = 0$
Set each factor to zero and solve for $x$ .
$x+5 = 0 \implies x = -5$ $x+10 = 0 \implies x = -10$
Check if these values make the denominator zero.
For $x = -5$ , the denominator is $-5 - 5 = -10 \neq 0$ .
For $x = -10$ , the denominator is $-10 - 5 = -15 \neq 0$ .
Since neither value makes the denominator zero, both are valid solutions.

The values of $x$ for which the expression is zero are $\boxed{\textbf{-5 \text{ and } -10}}$ .

For what value(s) of x is the expression racx2 + 15x + 50x-5 not defined? Find the value(s) of x for which the expression is zero.