How do you solve $5/x - 2 = 2/(x+3)$ ?

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Answer 1 · 2015-11-17T11:31:24

$x = -3/4 + sqrt(129)/4 color(white)(xx)$ and $color(white)(xx)x = -3/4 - sqrt(129)/4$

First of all, please be aware that $x != 0$ and $x != -3$ must hold so that the equation is defined.

Now, with these restrictions in mind, let's start solving. :)

First of all, we need to get rid of the denominators. To do so, multiply both sides with $x * (x+3)$ :

$color(white)(xxxss) 5/x - 2 = 2 / (x+3)$

$<=> color(white)(xx) 5 ( x + 3 ) - 2 x (x + 3) = 2 x$
$<=> color(white)(xx) 5x + 15 - 2x^2 - 6x = 2x$

... add $-2x$ on both sides and simplify ...

$<=> color(white)(xx) 15 - 3x - 2x^2 = 0$

This is a quadratic equation which can be solved for example with the quadratic formula which states that the solution of $ax^2 + bx + c = 0$ is

$x = (-b +- sqrt(b^2-4ac))/(2a)$

In our case, $a = -2$ , $b = -3$ and $c = 15$ .

This leads us to

$x = (3 +- sqrt(9 - 4 * (-2)*15))/-4 = (3 +- sqrt(129))/-4$

The two solutions are

$x = -3/4 + sqrt(129)/4 color(white)(xx)$ and $color(white)(xx)x = -3/4 - sqrt(129)/4$

How do you solve 5/x - 2 = 2/(x+3)5/x - 2 = 2/(x+3)?