1954 AMC 12 Problem 2

Question

The equation

\begin{aligned} \frac{2x^2}{x-1}-\frac{2x+7}{3} &{}+\frac{4-6x}{x-1}\\ &{}+1=0 \end{aligned}

can be transformed by eliminating fractions to the equation

x^2-5x+4=0.

The roots of the latter equation are

4

and

1.

Then the roots of the first equation are:

4

and

1

only

1

only

4

neither

4

nor

1

4

and some other root

Check both roots against the denominator in the original equation

Clearing a denominator can introduce a value at which the original expression is undefined

The transformed quadratic factors as

(x-1)(x-4)=0.

However, the original equation has denominator

x-1,

so it is undefined at

x=1.

Substitution of

x=4

is valid and satisfies the original equation.

Thus, the only root is

4,

and the correct answer is C.

Problem 2 in other years