Question

How do you multiply and simplify `\frac { x - 3} { 3x } \cdot \frac { 6} { x ^ { 2} - 9}`?

Answer 1

See a solution process below:

Explanation:

The numerator of the fraction on the right is the special quadratic form:

`(a + b)(a - b) = a^2 - b^2`

Using this rule we can rewrite the expression as:

`(x - 3)/(3x) * 6/(x^2 - 9) => (x - 3)/(3x) * 6/((x + 3)(x - 3))`

We can now cancel out common terms in the numerator and denominator:

`color(red)(cancel(color(black)(x - 3)))/(color(blue)(cancel(color(black)(3)))x) * (color(blue)(cancel(color(black)(6)))2)/((x + 3)color(red)(cancel(color(black)((x - 3))))) =>`

`2/(x(x + 3))`

Or

`2/(x^2 + 3x)`