How do extraneous solutions arise?

1 Answer
Mar 8, 2015

In general, extraneous solutions arise when we perform non-invertible operations on both sides of an equation. (That is, they sometimes arise, but not always.)

Non-invertible operations include: raising to an even power (odd powers are invertible), multiplying by zero, and combining sums and differences of logarithms.

Example :
The equations: x+2=9 and x=7, have exactly the same set of solutions. Namely: {7}.

Square both sides of x=7 to get the new equation: x2=49. The solution set of this new equation is; {7,7}. The 7 is an extraneous solution introduced by squaring the two expressions

Square both sides of x+2=9 to get the new equation: x2+4x+4=81. Solve the new equation:
x2+4x77=0 so (x7)(x+11)=0 whose solution set is {7,11}. The 11 does not solve the original equation.