Answer :
Let the two consecutive negative integers be [tex]$n$[/tex] and [tex]$n+1$[/tex]. The product of these integers is given by
[tex]$$
n(n+1) = 600.
$$[/tex]
Expanding the left side, we have
[tex]$$
n^2 + n = 600.
$$[/tex]
Subtracting 600 from both sides yields the quadratic equation:
[tex]$$
n^2 + n - 600 = 0.
$$[/tex]
For a quadratic equation of the form [tex]$ax^2+bx+c=0$[/tex], the discriminant is given by
[tex]$$
D = b^2 - 4ac.
$$[/tex]
Here, [tex]$a=1$[/tex], [tex]$b=1$[/tex], and [tex]$c=-600$[/tex]. Plugging in these values:
[tex]$$
D = 1^2 - 4(1)(-600) = 1 + 2400 = 2401.
$$[/tex]
Taking the square root of the discriminant:
[tex]$$
\sqrt{D} = \sqrt{2401} = 49.
$$[/tex]
Now, applying the quadratic formula
[tex]$$
n = \frac{-b \pm \sqrt{D}}{2a},
$$[/tex]
we find the two possible solutions:
[tex]$$
n = \frac{-1 + 49}{2} = \frac{48}{2} = 24
$$[/tex]
and
[tex]$$
n = \frac{-1 - 49}{2} = \frac{-50}{2} = -25.
$$[/tex]
Since the problem specifies consecutive negative integers, we select the negative solution. Therefore, the lesser integer is
[tex]$$
\boxed{-25}.
$$[/tex]
[tex]$$
n(n+1) = 600.
$$[/tex]
Expanding the left side, we have
[tex]$$
n^2 + n = 600.
$$[/tex]
Subtracting 600 from both sides yields the quadratic equation:
[tex]$$
n^2 + n - 600 = 0.
$$[/tex]
For a quadratic equation of the form [tex]$ax^2+bx+c=0$[/tex], the discriminant is given by
[tex]$$
D = b^2 - 4ac.
$$[/tex]
Here, [tex]$a=1$[/tex], [tex]$b=1$[/tex], and [tex]$c=-600$[/tex]. Plugging in these values:
[tex]$$
D = 1^2 - 4(1)(-600) = 1 + 2400 = 2401.
$$[/tex]
Taking the square root of the discriminant:
[tex]$$
\sqrt{D} = \sqrt{2401} = 49.
$$[/tex]
Now, applying the quadratic formula
[tex]$$
n = \frac{-b \pm \sqrt{D}}{2a},
$$[/tex]
we find the two possible solutions:
[tex]$$
n = \frac{-1 + 49}{2} = \frac{48}{2} = 24
$$[/tex]
and
[tex]$$
n = \frac{-1 - 49}{2} = \frac{-50}{2} = -25.
$$[/tex]
Since the problem specifies consecutive negative integers, we select the negative solution. Therefore, the lesser integer is
[tex]$$
\boxed{-25}.
$$[/tex]
