1950 AMC 12 Problem 38

Question

If the expression

\begin{vmatrix}a&c\\d&b\end{vmatrix}

has the value

ab-cd

for all values of

a,

b,

c,

and

d,

then the equation

\begin{vmatrix}2x&1\\x&x\end{vmatrix}=3

Is satisfied for only

1

value of

x

Is satisfied for

2

values of

x

Is satisfied for no values of

x

Is satisfied for an infinite number of values of

x

None of these

Apply the given rule to turn the determinant into a quadratic equation

The equation is

2x^2-x=3

The given rule turns the equation into

(2x)(x)-(1)(x)=3,

or

2x^2-x-3=0.

Factoring gives

(2x-3)(x+1)=0,

whose two distinct solutions are

x=\tfrac32

and

x=-1.

Thus, the correct answer is B.

Problem 38 in other years