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

1 Answer
Jun 11, 2015

Answer:

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.