A fractional equation may become quadratic after multiplying by a common denominator. State denominator restrictions first. Solving the new equation gives candidates; each must still satisfy the original restrictions and context. Example Solve 6x+1=x, where x!= -1. 6=x(x+1) x2+x-6=0 (x+3)(x-2)=0 x=-3 x=2. Both are permitted: 6-3+1=-3 and 62+1=2. A rectangle has width w cm, length w+3 cm and area 40 cm². Then w(w+3)=40, so (w+8)(w-5)=0. Only w=5 is a valid width; the length is 8 cm. Reject -8 because of the context, not because negative roots are always invalid.

A fractional equation may become quadratic after multiplying by a common denominator. State denominator restrictions first. Solving the new equation gives candidates; each must still satisfy the original restrictions and context.

Solve \(\dfrac6{x+1}=x\), where \(x\ne-1\).

\(\begin{aligned}6&=x(x+1)\\x^2+x-6&=0\\(x+3)(x-2)&=0\\x&=-3\quad\text{or}\quad x=2.\end{aligned}\)

Both are permitted: \(\dfrac6{-3+1}=-3\) and \(\dfrac6{2+1}=2\).

A rectangle has width \(w\) cm, length \(w+3\) cm and area \(40\) cm². Then \(w(w+3)=40\), so \((w+8)(w-5)=0\). Only \(w=5\) is a valid width; the length is \(8\) cm. Reject \(-8\) because of the context, not because negative roots are always invalid.

Fractional equations and admissible roots

Fractional equations and admissible roots