Let the two consecutive negative integers be \(x\) and \(x + 1\). The product of these integers can be expressed as:
\[ x(x + 1) = 600 \]
This simplifies to:
\[ x^2 + x - 600 = 0 \]
Now, we will solve this quadratic equation using the quadratic formula:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]
In our case, \(a = 1\), \(b = 1\), and \(c = -600\). Plugging these values into the formula, we get:
\[ x = \frac{{-1 \pm \sqrt{{1^2 - 4 \cdot 1 \cdot (-600)}}}}{2 \cdot 1} \]
Calculating the discriminant:
\[ b^2 - 4ac = 1 + 2400 = 2401 \]
Now, we can find the square root of \(2401\):
\[ \sqrt{2401} = 49 \]
Substituting this back into the equation gives:
\[ x = \frac{{-1 \pm 49}}{2} \]
Calculating the two possible values for \(x\):
- Using \(49\):
\[ x = \frac{{-1 + 49}}{2} = \frac{48}{2} = 24 \]
- Using \(-49\):
\[ x = \frac{{-1 - 49}}{2} = \frac{-50}{2} = -25 \]
Since we are looking for negative integers, we will take \(x = -25\).
Thus, the lesser integer is:
\[ \boxed{-25} \]
To confirm, the two consecutive integers are \(-25\) and \(-24\), and their product is:
\[ -25 \times -24 = 600 \]
This checks out, confirming that the lesser integer is indeed \(-25\).
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