Answer :
To solve the problem of finding the lesser of two consecutive negative integers whose product is 600, let's break it down step-by-step:
1. Define the Integers:
Let's call the two consecutive negative integers [tex]\( x \)[/tex] and [tex]\( x + 1 \)[/tex].
2. Set Up the Equation:
The problem states that their product is 600. So, we have the equation:
[tex]\[
x \times (x + 1) = 600
\][/tex]
3. Expand and Rearrange:
Distribute [tex]\( x \)[/tex] in the equation:
[tex]\[
x^2 + x = 600
\][/tex]
Subtract 600 from both sides to set the equation to zero:
[tex]\[
x^2 + x - 600 = 0
\][/tex]
4. Solve the Quadratic Equation:
The equation [tex]\( x^2 + x - 600 = 0 \)[/tex] is a quadratic equation and can be solved using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -600 \)[/tex].
5. Calculate the Discriminant:
The discriminant is given by [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[
1^2 - 4 \times 1 \times (-600) = 1 + 2400 = 2401
\][/tex]
6. Find the Roots:
Use the quadratic formula to find [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-1 \pm \sqrt{2401}}{2}
\][/tex]
Simplifying gives us two potential solutions for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-1 + 49}{2} = 24
\][/tex]
[tex]\[
x = \frac{-1 - 49}{2} = -25
\][/tex]
7. Choose the Negative Integer:
We are looking for negative integers, so the solution that fits is [tex]\( x = -25 \)[/tex].
Thus, the lesser of the two consecutive negative integers is [tex]\(-25\)[/tex].
1. Define the Integers:
Let's call the two consecutive negative integers [tex]\( x \)[/tex] and [tex]\( x + 1 \)[/tex].
2. Set Up the Equation:
The problem states that their product is 600. So, we have the equation:
[tex]\[
x \times (x + 1) = 600
\][/tex]
3. Expand and Rearrange:
Distribute [tex]\( x \)[/tex] in the equation:
[tex]\[
x^2 + x = 600
\][/tex]
Subtract 600 from both sides to set the equation to zero:
[tex]\[
x^2 + x - 600 = 0
\][/tex]
4. Solve the Quadratic Equation:
The equation [tex]\( x^2 + x - 600 = 0 \)[/tex] is a quadratic equation and can be solved using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -600 \)[/tex].
5. Calculate the Discriminant:
The discriminant is given by [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[
1^2 - 4 \times 1 \times (-600) = 1 + 2400 = 2401
\][/tex]
6. Find the Roots:
Use the quadratic formula to find [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-1 \pm \sqrt{2401}}{2}
\][/tex]
Simplifying gives us two potential solutions for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-1 + 49}{2} = 24
\][/tex]
[tex]\[
x = \frac{-1 - 49}{2} = -25
\][/tex]
7. Choose the Negative Integer:
We are looking for negative integers, so the solution that fits is [tex]\( x = -25 \)[/tex].
Thus, the lesser of the two consecutive negative integers is [tex]\(-25\)[/tex].
