Question

Find all the possible of x for which of the below is true:


`(2x)/(x + 3) + (3)/(x - 3) = (8)/(x^(2) - 9 )`
​

Answer 1

x =
`(1)/(2)` , x = 1

Explanation:

Given

`(2x)/(x+3)` +
`(3)/(x-3)` =
`(8)/(x^2-9)` ← denominator is a difference of squares

`(2x)/(x+3)` +
`(3)/(x-3)` =
`(8)/((x+3)(x-3))`

Multiply through by (x + 3)(x - 3) to clear the fractions

2x(x - 3) + 3(x + 3) = 8 ← distribute left side and simplify

2x² - 6x + 3x + 9 = 8

2x² - 3x + 9 = 8 ( subtract 8 from both sides )

2x² - 3x + 1 = 0 ← in standard form

(2x - 1)(x - 1) = 0 ← in factored form

Equate each factor to zero and solve for x

2x - 1 = 0 ⇒ 2x = 1 ⇒ x =
`(1)/(2)`

x - 1 = 0 ⇒ x = 1