Question

How do you solve the rational equation `(x^2-7)/(x-4)=0`?

Answer 1

`x = ±sqrt7`

Explanation:

for this rational function to equate to zero means that the numerator must be zero as the denominator ≠ 0

`rArr x^2 - 7 = 0 rArr x^2 = 7 rArr x = ±sqrt7`

Answer 2

`x=+-sqrt(7)`

Strong guidance given on manipulation!

Explanation:

Given: `color(brown)(color(white)(....) (x^2-7)/(x-4)=0)`

Multiply both sides by `color(blue)((x-4))`

`color(brown)((x^2-7)/(x-4)color(blue)(xx(x-4))=0xxcolor(blue)((x-4))`

`(x^2-7) xx(x-4)/(x-4)=0`

But `(x-4)/(x-4)" is another way of writing " 1` giving:

`(x^2-7) xx1=0`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Add `color(blue)(7)` to both sides

`color(brown)( x^2-7 color(blue)(+7) =0color(blue)(+7)`

`x^2=7`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

so `color(white)(...) x= sqrt(x^2) = +-sqrt(7)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The square root has to be `+-` because:

`x^2=( + sqrt(7))xx (+sqrt(7) )-> "positive "x^2`

And

`x^2= (-sqrt(7))xx(-sqrt(7))-> "positive "x^2`