Question

When solving a rational equation, why is it necessary to perform a check?

Answer 1

It is necessary to perform a check because in the process of multiplying through you can introduce spurious solutions.

Explanation:

Consider the example:

`(x+3)/(x^2-3x+2) = (x+2)/(x^2-4x+3)`

We could choose to "cross multiply" the equation to get:

`(x+3)(x^2-4x+3) = (x+2)(x^2-3x+2)`

That is:

`x^3-x^2-9x+9=x^3-x^2-4x+4`

Subtract `x^3-x^2` from both sides to get:

`-9x+9 = -4x+4`

Add `4x-4` to both sides to get:

`-5x+5 = 0`

Divide both sides by `5` to get

`-x+1 = 0`

Hence `x = 1`

But try putting `x=1` in the original equation and you will find both denominators are zero.

What went wrong here is that both `(x^2-3x+2)` and `(x^2-4x+3)` are divisible by `(x-1)`, so cross multiplying by them included the effect of multiplying both sides by `(x-1)^2` - not only clearing `(x-1)` from the denominator, but adding an extra factor of `(x-1)` on both sides of the equation.