What is the common denominator

Question

Help me find the common denominator please!

Thank you

Here it is :

$$\dfrac{5}{3x+2}-\dfrac{3}{3x-2}=\dfrac{7}{6x-4}+\dfrac{x+4}{18x^2-8}. 1$$

Answer 1

Be sure that you need to check all factors in the denominator and then, determine the least common denominator that is divisible by all known denominators. That is...

Observe that $6x - 4 = 2(3x - 2)$ and $18x^2 - 8 = 2(9x^2 - 4) = 2(3x - 2)(3x + 2)$. Then, the least common denominator of that expression is $2(3x + 2)(3x - 2)$; $3x - 2$, $6x - 4$ and $18x^2 - 8$ have the common factor of $3x - 2$, whereas $18x^2 - 8$ and $3x + 2$ has the common factor of $3x + 2$.

Answer 2

To find common denominators, factoring usually gives you the result. This equation is no exception.

$18x^2-8$ can be factored as $2(3x+2)(3x-2)$.

$6x-4$ can be factored as $2(3x-2)$. Multiplying $6x-4$ by $3x+2$ gives you $18x^2-8$.

$3x+2$ is not factorable. Multiplying it by $2(3x-2)$ gives you $18x^2-8$.

$3x-2$ is not factorable. Multiplying it by $2(3x+2)$ gives you $18x^2-8$.

Therefore $18x^2-8$ is the least common multiple.