Question

table of contents > module 7: rational expressions and equations > practice: add and subtract rat
practice: add and subtract rational
part i
practice: add and subtract rational expressions part i
score: 4/6 answered: 4/6
question 5
add \\(\frac{4x}{x^2 - 4x + 4} + \frac{5}{x - 2}\\) and state the sum in simplest form.
the domain is \\(x \
eq\\)

Explanation:

Step1: Factor the denominator

Factor \(x^2 - 4x + 4\). We know that \(x^2 - 4x + 4=(x - 2)^2\) (using the formula \((a - b)^2=a^2-2ab + b^2\) where \(a = x\) and \(b = 2\)). So the first rational expression becomes \(\frac{4x}{(x - 2)^2}\) and the second is \(\frac{5}{x - 2}\).

Step2: Find a common denominator

The common denominator of \((x - 2)^2\) and \(x - 2\) is \((x - 2)^2\). Rewrite the second fraction with the common denominator: \(\frac{5}{x - 2}=\frac{5(x - 2)}{(x - 2)^2}\) (multiply numerator and denominator by \(x - 2\)).

Step3: Add the fractions

Now add the two fractions: \(\frac{4x}{(x - 2)^2}+\frac{5(x - 2)}{(x - 2)^2}=\frac{4x+5(x - 2)}{(x - 2)^2}\). Expand the numerator: \(4x + 5x-10=9x - 10\). So the sum is \(\frac{9x - 10}{(x - 2)^2}\) (we can check if this can be simplified further, and since the numerator \(9x - 10\) and denominator \((x - 2)^2\) have no common factors, this is the simplest form).

Step4: Determine the domain

For the domain, we need to find the values of \(x\) that make the denominators zero. The denominators are \((x - 2)^2\) and \(x - 2\). Setting \(x - 2 = 0\) gives \(x = 2\). So the domain is \(x
eq2\).

Answer:

The sum in simplest form is \(\frac{9x - 10}{(x - 2)^2}\) and the domain is \(x
eq2\).