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Answer :
Let the two consecutive negative integers be [tex]$n$[/tex] and [tex]$n+1$[/tex]. Their product is given by
[tex]$$
n(n+1) = 600.
$$[/tex]
Expanding the left side, we obtain
[tex]$$
n^2 + n = 600.
$$[/tex]
To form a quadratic equation, subtract [tex]$600$[/tex] from both sides:
[tex]$$
n^2 + n - 600 = 0.
$$[/tex]
Now, we will solve this quadratic equation using the quadratic formula:
[tex]$$
n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
$$[/tex]
where [tex]$a=1$[/tex], [tex]$b=1$[/tex], and [tex]$c=-600$[/tex]. First, compute the discriminant:
[tex]$$
\Delta = b^2 - 4ac = 1^2 - 4(1)(-600)= 1 + 2400 = 2401.
$$[/tex]
Taking the square root of [tex]$\Delta$[/tex],
[tex]$$
\sqrt{2401} = 49.
$$[/tex]
Now substitute into the quadratic formula:
[tex]$$
n = \frac{-1 \pm 49}{2}.
$$[/tex]
This gives two solutions:
1.
[tex]$$
n = \frac{-1 + 49}{2} = \frac{48}{2} = 24,
$$[/tex]
2.
[tex]$$
n = \frac{-1 - 49}{2} = \frac{-50}{2} = -25.
$$[/tex]
Since we are looking for two consecutive negative integers, we select the negative solution. Therefore, the two integers are [tex]$-25$[/tex] and [tex]$-24$[/tex].
Thus, the value of the lesser integer is
[tex]$$
\boxed{-25}.
$$[/tex]
[tex]$$
n(n+1) = 600.
$$[/tex]
Expanding the left side, we obtain
[tex]$$
n^2 + n = 600.
$$[/tex]
To form a quadratic equation, subtract [tex]$600$[/tex] from both sides:
[tex]$$
n^2 + n - 600 = 0.
$$[/tex]
Now, we will solve this quadratic equation using the quadratic formula:
[tex]$$
n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
$$[/tex]
where [tex]$a=1$[/tex], [tex]$b=1$[/tex], and [tex]$c=-600$[/tex]. First, compute the discriminant:
[tex]$$
\Delta = b^2 - 4ac = 1^2 - 4(1)(-600)= 1 + 2400 = 2401.
$$[/tex]
Taking the square root of [tex]$\Delta$[/tex],
[tex]$$
\sqrt{2401} = 49.
$$[/tex]
Now substitute into the quadratic formula:
[tex]$$
n = \frac{-1 \pm 49}{2}.
$$[/tex]
This gives two solutions:
1.
[tex]$$
n = \frac{-1 + 49}{2} = \frac{48}{2} = 24,
$$[/tex]
2.
[tex]$$
n = \frac{-1 - 49}{2} = \frac{-50}{2} = -25.
$$[/tex]
Since we are looking for two consecutive negative integers, we select the negative solution. Therefore, the two integers are [tex]$-25$[/tex] and [tex]$-24$[/tex].
Thus, the value of the lesser integer is
[tex]$$
\boxed{-25}.
$$[/tex]
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