Let's determine the real value of $m$ such that the roots of the equation

$$
x^{2}-(m-1) x+m+2=0
$$

are real.

Plot the real roots $x_{1}$ and $x_{2}$ on the x-axis, and draw circles centered at these points that pass through the origin.

a) What condition must $m$ satisfy so that these two circles are externally tangent?
b) For what value of $m$ will the radii of the two externally tangent circles be equal?

c) Determine the length of the segment defined by the points of tangency on the common tangent of the two circles as a function of $m$.
The roots of the quadratic equation are real if the discriminant is non-negative, that is, if

$$
(m-1)^{2}-4(m+2)=m^{2}-6 m-7=(m-7)(m+1) \geq 0
$$

This is clearly satisfied if both factors are simultaneously of the same sign. Both roots are positive if \( m > 7 \), and both roots are negative if \( m < -1 \). For \( -1 < m < 7 \), the equation has no real roots.

a) The two circles touch externally if one root is positive and the other is negative. In this case, the product of the roots is negative, which means the constant term

$$
m+2<0, \quad \text { that is } \quad m<-2
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_7790651734ff4cb75355g-1.jpg?height=529&width=826&top_left_y=592&top_left_x=625)

(In the boundary case \( m = -2 \), one of the circles shrinks to a point.)

b) The radii of the two circles are equal if

$$
x_{1}=-x_{2},
$$

which means

$$
x_{1}+x_{2}=m-1=0, \quad \text { that is } \quad m=1
$$

but in this case, as we have seen, there are no real roots.

Therefore, the radii of the two circles cannot be equal.

c) We can only speak of a common tangent segment if the two circles touch externally, that is, if \( m < -2 \). From the right triangle \( O_{1} E_{2}^{\prime} O_{2} \) (see the figure),

$$
d^{2}=\left(x_{1}+\left|x_{2}\right|\right)^{2}-\left(\left|x_{2}\right|-x_{1}\right)^{2}=4 x_{1}\left|x_{2}\right|=4(-m-2)
$$

since \( x_{1} x_{2}=m+2 \) is negative.

Thus,

$$
\begin{aligned}
& d=2 \sqrt{-m-2} \text {. } \\
& \quad \text { Jajczay Ágnes (Bp., IX. Patrona Hungariae lg. III. o. t.) }
\end{aligned}
$$