The best way to do this is to use your LCM and eliminate the fractions. To find the LCM you have to use all the denominators as a multiplier so the denominator in each term cancels out. We will first factor the x-squared term to simplify and see what 2 factors are hidden there.

$x^2-4$ factors to (x + 2)(x - 2). That means that our 3 denominators that make up our LCM are x(x+2)(x-2). We will mulitply that in to each term in our rational equation, canceling out the denominators where applicable.

$x(x+2)(x-2)[\frac{2}{(x-2)}+\frac{7}{(x-2)(x+2)}=\frac{5}{x}]$

In the first term, the (x-2) will cancel leaving us with

x(x+2)[2] which simplifies to

$x^2+2x[2]$

In the second term, the (x+2)(x-2) cancels out leaving us with

x[7].

In the last term, the x cancels out leaving us with

(x+2)(x-2)[5] which simplifies to

$x^2-4[5]$

Now we will distribute through each cancellation:

2x²+4x;

7x;

5x²-20

Putting them all together we have

2x² + 4x + 7x = 5x² - 20

Combining like terms gives us a quadratic:

3x² - 11x - 20 = 0

Factor that however you find it easiest to factor quadratics and get that

x = 5 and x = -4/3

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3 years ago