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Chapter 0: Problem 6

Find all numbers that must be excluded from the domain of each rational expression. $$\frac{x-3}{x^{2}+4 x-45}$$

Short Answer

Expert verified
The numbers that must be excluded from the domain of the rational expression \(\frac{x-3}{x^{2}+4 x-45}\) are \(x = 5\) and \(x = -9\). These are the numbers that would make the denominator equal to zero and thus make the expression undefined.

Step by step solution

01

Write down the expression

The given rational expression is \(\frac{x-3}{x^{2}+4 x-45}\)
02

Set the denominator equal to 0 and solve for x

To find out the values which we need to exclude from the domain (i.e. values which will make the denominator zero), we set the denominator equal to zero and solve for x. So, \(x^{2}+4x-45 = 0\)
03

Factorize the quadratic expression

This quadratic expression can be factored into two binomial expressions. Let's factorize \(x^{2}+4x-45\) to find the roots. This is done by finding two numbers that add up to 4 (the coefficient of x) and multiply to -45 (the constant term). The numbers that fit these conditions are 9 and -5. Hence, our quadratic equation will factor to \((x-5)(x+9) = 0\)
04

Set each factor equal to zero to solve for x

We set each factor equal to zero and solve for x: \(x - 5 = 0\) will give \(x = 5\) and \(x + 9 = 0\) will give \(x = -9\)

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