1956 AMC 12 Problem 23

Question

About the equation

ax^2-2x\sqrt2+c=0,

with

a

and

c

real constants, we are told that the discriminant is zero. The roots are necessarily:

equal and integral

equal and rational

equal and real

equal and irrational

equal and imaginary

A zero discriminant makes the two quadratic-formula values coincide

The repeated root is

-\frac{b}{2a}=\frac{\sqrt2}{a}

A quadratic with real coefficients and discriminant zero has the repeated root

x=\frac{-(-2\sqrt2)}{2a}=\frac{\sqrt2}{a}.

Since

a

is a nonzero real number, this root is real. It need not always be rational or always be irrational.

Thus, the roots are necessarily equal and real, so the correct answer is C.

Problem 23 in other years