Question

The product of two consecutive negative integers is 600. What is the value of the lesser integer?

–60
–30
–25
–15

Answer 1

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 \(x\):

1. Using \(49\):

\[
x = \frac{{-1 + 49}}{2} = \frac{48}{2} = 24
\]

2. 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 \(-25\).