Question

the product of two consecutive negative integers is 600. what is the value of the lesser integer?
-60
-30
-25
-15

Explanation:

Step1: Define variables

Let the lesser negative integer be \( x \), then the next consecutive negative integer is \( x + 1 \) (since consecutive integers differ by 1).

Step2: Set up equation

The product of the two consecutive negative integers is 600, so we have the equation:

$$ x(x + 1)=600 $$

Expanding this gives:

$$ x^{2}+x - 600 = 0 $$

Step3: Solve the quadratic equation

We can factor the quadratic equation. We need two numbers that multiply to - 600 and add up to 1. The numbers are 25 and - 24? Wait, no. Wait, let's try to factor \(x^{2}+x - 600\). We can also use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for the quadratic equation \(ax^{2}+bx + c = 0\). Here, \(a = 1\), \(b=1\), \(c=-600\).

First, calculate the discriminant \(\Delta=b^{2}-4ac=(1)^{2}-4\times1\times(-600)=1 + 2400=2401\)

Then, \(x=\frac{-1\pm\sqrt{2401}}{2}=\frac{-1\pm49}{2}\)

We have two solutions:
\(x_{1}=\frac{-1 + 49}{2}=\frac{48}{2}=24\) and \(x_{2}=\frac{-1-49}{2}=\frac{-50}{2}=-25\)

Since we are looking for negative integers, we consider \(x=-25\). Let's check: the next consecutive integer is \(-25 + 1=-24\). The product of \(-25\) and \(-24\) is \((-25)\times(-24)=600\), which matches the given condition.

We can also check the options:

• For - 60: The next integer is - 59, \((-60)\times(-59)=3540

eq600\)

• For - 30: The next integer is - 29, \((-30)\times(-29)=870

eq600\)

• For - 25: The next integer is - 24, \((-25)\times(-24)=600\) (correct)
• For - 15: The next integer is - 14, \((-15)\times(-14)=210

eq600\)

Answer:

-25